Quantum Spin-Hall systems are topological insulators displaying dissipationless spin currents flowing at the edges of the samples. In contradistinction to the Quantum Hall systems where the charge conductance of the edge modes is quantized, the spin conductance is not and it remained an open problem to find the observable whose edge current is quantized. In this paper, we define a particular observable and the edge current corresponding to this observable. We show that this current is quantized and that the quantization is given by the index of a certain Fredholm operator. This provides a new topological invariant that is shown to take same values as the Spin-Chern number previously introduced in the literature. The result gives an effective tool for the investigation of the edge channels' structure in Quantum Spin-Hall systems. Based on a reasonable assumption, we also show that the edge conducting channels are not destroyed by a random edge.
On the surface of 2D Topological insulators occur 1D quantum spin Hall(QSH) edge modes with Dirac like dispersion. Unlike quantum Hall(QH) edge modes which occur at high magnetic fields in 2DEGs, the occurrence of QSH edge modes is because of spin-orbit scattering in the bulk of the material. These QSH edge modes are spin dependent and chiral- opposite spins move in opposing directions. Electronic spin has larger decoherence and relaxation time than charge- in view of this its expected that QSH edge modes will be more robust to disorder and inelastic scattering than QH edge modes which are cha…
Properties of eigenstates of one-particle Quantum Hall Hamiltonians localized near the boundary of a two-dimensional electron gas - so-called edge states - are studied. For finite samples it is shown that edge states with energy in an appropriate range between Landau levels remain extended along the boundary in the presence of a small amount of disorder, in the sense that they carry a non-zero chiral edge current. For a two-dimensional electron gas confined to a half-plane, the Mourre theory of positive commutators is applied to prove absolute continuity of the energy spectrum well in between …
We propose a quasi-particle formulation of effective edge theories for the fractional quantum Hall effect. For the edge of a Laughlin state with filling fraction ν=1/m, our fundamental quasi-particles are edge electrons of charge -e and edge quasi-holes of charge +e/m. These quasi-particles satisfy exclusion statistics in the sense of Haldane. We exploit algebraic properties of edge electrons to derive a kinetic equation for charge transport between a ν=1/m fractional quantum Hall edge and a normal metal. We also analyze alternative `Boltzmann' equations that are directly based on the exclusio…
We calculate numerically the localization length critical index within the Chalker-Coddington (CC) model for plateau-plateau transitions in the quantum Hall effect. Lyapunov exponents have been calculated with relative errors on the order $10^{-3}$. Such high precision was obtained by considering the distribution of Lyapunov exponents for large ensembles of relatively short chains and calculating the ensemble average values. We analyze thoroughly finite size effects and find the localization length critical index $ν= 2.517\pm 0.018$.
We examine the influence of the edge electronic density profile and of dissipation on edge magnetoplasmons in the quantum Hall regime, in a semiclassical calculation. The equilibrium electron density on the edge, obtained using a Thomas-Fermi approach, has incompressible stripes produced by energy gaps responsible for the quantum Hall effect. We find that these stripes have an unobservably small effect on the edge magnetoplasmons. But dissipation, included phenomenologically in the local conductivity, proves to produce significant oscillations in the strength and speed of edge magnetoplasmons …
This is a joint publication with the Letter by H. Huang and F. Liu [Phys. Rev. Lett. 121, 126401 (2018)]. In this work, we propose the spin Bott index to identify the quantum spin Hall (QSH) state in both crystalline and non-periodic systems. The applicability of the spin Bott index is confirmed by analyzing the periodic and disorder Kane-Mele models. As an example of non-periodic systems, we systematically investigate the QSH effect in a Penrose-type quasicrystal lattice (QL). We characterized the nontrivial electronic topology of the QL by directly calculating the spin Bott index. In additio…
In the paper we dscuss experimental results of M. Grayson et al. on tunneling $I$-$V$ characteristics of the quantum Hall edge. We suggest a two step tunneling mechanism involving localized electron states near the edge, which might account for discrepancy between the experimental data and the predictions of the chiral Luttinger liquid theory of the quantum Hall edge.
We study the interplay of interaction, confining potential and effects of finite temperature at the edge of a quantum Hall liquid. Our exact diagonalization calculation indicates that edge reconstruction occurs in the fractional quantum Hall regime for a variety of confining potential, including ones that correspond to a "sharp" edge. Our finite temperature Hartree-Fock calculation for integer quantum Hall edges indicates that reconstruction is suppressed above certain temperature. We discuss the implication of our results on recent edge tunneling and microwave absorption experiments.
Two dimensional electron systems exhibiting the fractional quantum Hall effects are characterized by a quantized Hall conductance and a dissipationless bulk. The transport in these systems occurs only at the edges where gapless excitations are present. We present a {\it microscopic} calculation of the edge states in the fractional quantum Hall systems at various filling factors using the extended Hamiltonian theory of the fractional quantum Hall effect. We find that at $ν=1/3$ the quantum Hall edge undergoes a reconstruction as the background potential softens, whereas quantum Hall edges at hi…
Employing the Bloch eigenmode matching approach, we numerically study the evolution of individual quantum Hall edge states with respect to disorder. As shown by the two-parameter renormalization group flow of the Hall and Thouless conductances, quantum Hall edge states with high Chern number n are completely different from that of n=1 case. Two categories of individual edge modes are evaluated in a quantum Hall system with high Chern number. Edge states from the lowest Landau level have similar eigenfunctions which are well localized at the system edge and independent of the Fermi energy. On t…
We show that the Chern-Simons-Landau-Ginsburg theory that describes the quantum Hall effect on a bounded sample is anomaly free and thus does not require the addition of extra chiral fermions on the boundary to restore local gauge invariance.
We discuss a quasi-particle formulation of effective edge theories for the fractional quantum Hall effect. Fundamental quasi-particles for the Laughlin state with filling fraction ν=1/3 are edge electrons of charge -e and edge quasi-holes of charge +e/3. These quasi-particles satisfy exclusion statistics in the sense of Haldane. We exploit algebraic properties of edge electrons to derive a kinetic equation for charge transport between a ν=1/3 fractional quantum Hall edge and a normal metal.
The interplay between the confinement potential and electron-electron interactions causes reconstructions of Quantum Hall edges. We study the consequences of this edge reconstruction for the relaxation of hot electrons injected into integer quantum Hall edge states. In translationally invariant edges, the relaxation of hot electrons is governed by three-body collisions which are sensitive to the electron dispersion and thus to reconstruction effects. We show that the relaxation rates are significantly altered in different reconstruction scenarios.
Stationary solutions of the Chern-Simons effective field theory for the fractional quantum Hall systems with edges are presented for Hall bar, disk and annulus. In the infinitely long Hall bar geometry (non compact case), the charge density is shown to be monotonic inside the sample. In sharp contrast, spatial oscillatory modes of charge density are found for the two circular geometries, which indicate that in systems with compact geometry, charge and current exist also far from the edges.
Charge equilibration between quantum-Hall edge states can be studied to reveal geometric structure of edge channels not only in the integer quantum Hall (IQH) regime but also in the fractional quantum Hall (FQH) regime particularly for hole-conjugate states. Here we report on a systematic study of charge equilibration in both IQH and FQH regimes by using a generalized Hall bar, in which a quantum Hall state is nested in another quantum Hall state with different Landau filling factors. This provides a feasible way to evaluate equilibration in various conditions even in the presence of scatterin…
The sharp ν=1 quantum Hall edge present for hard confinement is shown to have two modes that go soft as the confining potential softens. This signals a second order transition to a reconstructed edge that is either a depolarized spin-texture edge or a polarized charge density wave edge.
A Fabry-Perot-type interferometer is experimentally realized for electrons in a semiconductor device. A special experimental geometry creates interference conditions for co-propagating electrons in quantum Hall edge states, which results in oscillations of the current through the device. The visibility of these oscillations is found to increase at the high-field edge of the quantum Hall plateau.
The magneto- and Hall resistance of a locally gated cross shaped graphene Hall bar is calculated. The edge of the top gate is placed diagonally across the center of the Hall cross. Four-probe resistance is calculated using the Landauer-Büttiker formalism, while the transmission coefficients are obtained using the non-equilibrium Green's function approach. The interplay between transport due to edge channels and snake states is investigated. When two edge channels are occupied we predict oscillations in the Hall and the bend resistance as function of the magnetic field which are a consequence o…
We study the softening of the edge modes of the compact ferromagnetic v=1 Quantum Hall edge as the strength of the confining potential decreases. To obtain a realistic potential we use a displaced background charge plane. We find a phase transition into a spin textured edge for g<0.012 (g is the ratio betwen the Zeeman and Coulomb energies).
We study a $ν=1$ topological system with one twisting edge-state band and one normal edge-state band. For the twisting edge-state band, Fermi energy goes through the band three times, thus, having three edge states on one side of the sample; while the normal edge band contributes only one edge state on the other side of the sample. In such a system, we show that it consists of both topologically protected and unprotected edge states, and as a consequence, its Hall resistance depends on the location where the Hall measurement is done even for a translationally invariant system. This unique prop…
Can Majorana zero modes in quantum Hall edges survive edge reconstruction?
Aix Marseille Univ, Université de Toulon, CNRS, CPT（艾克斯马赛大学，土伦大学，法国国家科学研究中心，理论物理中心）；Department of Physics, Indian Institute of Science Education and Research (IISER) Kolkata（印度科学与教育研究所加尔各答分校物理系）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）
Hole-conjugate states of the fractional quantum Hall effect host counter-propagating 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 ma…
For two dimensional Schroedinger Hamiltonians we formulate boundary conditions that split the Hilbert space according to the chirality of the eigenstates on the boundary. With magnetic fields, and in particular, for Quantum Hall Systems, this splitting corresponds to edge and bulk states. Applications to the integer and fractional Hall effect and some open problems are described.
The edge Hall conductivity is shown to be an integer multiple of $e^2/h$ which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K-theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few percent of the total current can be carried by edge states.
Using a low-temperature atomic force microscope as a local voltmeter, we measure the Hall voltage profile in a quantum Hall conductor in the presence of a gate-induced non-equilibrium edge state population at n=3. We observe sharp voltage drops at the sample edges which are suppressed by re-equilibrating the edge states.
We address the issue of an apparent disagreement between theory and experiment on the I-V characteristic of electron tunneling from a metal to the edge of a two-dimensional electron gas in the fractional quantum Hall regime. A part of our result is a theory for the edge of the half-filled Landau level.
The edge states of the quantum Hall and fractional quantum Hall effect of a two-dimensional electron gas provide key access to the excitations of the bulk. Here we demonstrate controlled transmission of edge states in bilayer graphene. Controlled by an independent gate, the backscattering rate of the edge modes is continuously tuned from 0 to close to 1, with fractional quantized values corresponding to the sequential complete backscattering of individual modes. Our experiments demonstrate the feasibility to controllably manipulate edge states in bilayer graphene, thus opening the door to more…
Spin-polarized reconstruction of the v=1 quantum Hall edge is accompanied by a spatial modulation of the charge density along the edge. We find that this is also the case for finite quantum Hall droplets: current spin density functional calculations show that the so-called Chamon-Wen edge forms a ring of apparently localized electrons around the maximum density droplet (MDD). The boundaries of these different phases qualitatively agree with recent experiments. For very soft confinement, Chern-Simons Ginzburg-Landau theory indicates formation of a non-translational invariant edge with vortices …
The chiral Luttinger model for the edges of the fractional quantum Hall effect is obtained as the low energy limit of the Chern-Simons theory for the two dimensional system. In particular we recover the Kac-Moody algebra for the creation and annihilation operators of the edge density waves and the bosonization formula for the electronic operator at the edge.
Momentum resolved magneto-tunnelling spectroscopy is performed at a single sharp quantum Hall edge. We directly probe the structure of individual integer quantum Hall (QH) edge modes, and find that an epitaxially overgrown cleaved edge realizes the sharp edge limit, where the Chklovskii picture relevant for soft etched or gated edges is no longer valid. The Fermi wavevector in the probe quantum well probes the real-space position of the QH edge modes, and reveals inter-channel distances smaller than both the magnetic length and the Bohr radius. We quantitatively describe the lineshape of princ…
Tunneling from a two-dimensional contact into quantum-Hall edges is considered theoretically for a case where the barrier is extended, uniform, and parallel to the edge. In contrast to previously realized tunneling geometries, details of the microscopic edge structure are exhibited directly in the voltage and magnetic-field dependence of the differential tunneling conductance. In particular, it is possible to measure the dispersion of the edge-magnetoplasmon mode, and the existence of additional, sometimes counterpropagating, edge-excitation branches could be detected.
We study theoretically the two interacting one-dimensional helical modes at the edges of the quantum spin Hall systems. A new type of inter-edge correlated liquid (IECL) without the spin gap is found. This liquid shows the diverging density wave (DW) and superconductivity (SC) correlations much stronger than those of the spinfull electrons. Possible experimental observations are also discussed.
Time-reversal invariance places strong constraints on the properties of the quantum spin Hall edge. One such restriction is the inevitability of dissipation in a Josephson junction between two superconductors formed on such an edge without the presence of interaction. Interactions and spin-conservation breaking are key ingredients for the realization of dissipationless ac Josephson effect on such quantum spin Hall edges. We present a simple quantum impurity model that allows to create a dissipationless fractional Josephson effect on a quantum spin Hall edge. We then use this model to substanti…
Recent calculations of the edge tunneling exponents in quantum Hall states appear to contradict their topological nature. We revisit this issue and find no fundamental discrepancies. In a microscopic model of fractional quantum Hall liquids with electron-electron interaction and confinement, we calculate the edge Green's function via exact diagonalization. Our results for $ν= 1/3$ and 2/3 suggest that in the presence of Coulomb interaction, the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, are the parameters that are relevant to …
Strained graphene with lattice deformations has been demonstrated to give rise to large pseudomagnetic fields and host many exotic properties. Here, we propose a non-magnetic approach to realize a momentum-dependent out-of-plane spin splitting in strained graphene nanoribbons with a moderate spin-orbit coupling. This unique spin splitting distincts from the well-known Zeeman-type spin splitting and the Rashba-type spin splitting. Our analysis indicates that the competition between quantum spin Hall edge states and valley edge states in the nanoribbon leads to the unique spin splitting. The qua…
By using observations from pump-probe stroboscopic confocal microscopy and spectroscopy, we demonstrate the dynamics of trions and the fractional quantum Hall edge on the order of $\sim1$ ps. The propagation of the quantum Hall edge state excited by a voltage pulse is detected as a temporal change in reflectance in the downstream edge probed by optical pulses synchronized with the voltage pulse. The temporal resolution of such stroboscopic pump-probe measurements is as fast as the duration time of the probe pulse ($\sim1$ ps). This ultra-fast stroboscope measurement enables us to distinguish b…
We investigate how a magnetic field induces one-dimensional edge channels when the two-dimensional surface states of three-dimensional topological insulators become gapped. The Hall effect, measured by contacting those channels, remains quantized even in situations where the θ-term in the bulk and the associated surface Hall conductivities, σ_{xy}^S, are not quantized due to the breaking of time-reversal symmetry. The quantization arises as the θ-term changes by \pm 2 πn along a loop around n edge channels. Model calculations show how an interplay of orbital and Zeeman effects leads to quantum…
We discuss the implementation of a beam splitter for electron waves in a quantum Hall bilayer. Our architecture exploits inter-layer tunneling to mix edge states belonging to different layers. We discuss the basic working principle of the proposed coherent edge mixer, possible interferometric implementations based on existing semiconductor-heterojunction technologies, and advantages with respect to canonical quantum Hall interferometers based on quantum point contacts.
We present direct experimental evidence for nonlocal transport in HgTe quantum wells in the quantum spin Hall regime, in the absence of any external magnetic field. The data conclusively show that the non-dissipative quantum transport occurs through edge channels, while the contacts lead to equilibration between the counter-propagating spin states at the edge. We show that the experimental data agree quantitatively with the theory of the quantum spin Hall effect.
Wen's chiral Tomonaga-Luttinger model for the edge of an m-layer quantum Hall system of total filling factor nu=m/(pm +- 1) with even p, is derived as a random-phase approximation of the Chern-Simons theory for these states. The theory allows for a description of edges both in and out of equilibrium, including their collective excitation spectrum and the tunneling exponent into the edge. While the tunneling exponent is insensitive to the details of a nu=m/(pm + 1) edge, it tends to decrease when a nu=m/(pm - 1) edge is taken out of equilibrium. The applicability of the theory to fractional qua…
We consider the multiple edge states of the Laughlin state and the Pfaffian state. These edge states are globally constrained through the operator algebra of conformal field theory in the bulk. We analyze these constraints by introducing an expression of quantum hall state by the chiral vertex operators and obtain the multiple edge partition functions by using the Verlinde formula.
Hall and diagonal resistances of bilayer fractional quantum Hall systems are discussed theoretically. The bilayers have electrodes attached separately to each layer. They are assumed to be coupled weakly by interlayer tunneling, while the interlayer Coulomb interaction is negligibly small. It is shown that source-drain voltage dependence of the resistances reflects the Luttinger liquid parameter of the edge state.
We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant Umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. These gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological eff…
We use a novel sample geometry to study non-local effects of edge excitations in the integer quantum Hall effect regime. We find that the condition of local equilibrium at the quantum Hall edge is affected by the diffusion of dynamically polarized nuclei. Our analysis indicates, that the nuclear diffusion is effectively one-dimensional in the present experiment.
We experimentally study equilibration between edge states, co-propagating at the edge of the fractional quantum Hall liquid, at high initial imbalances. We find an anomalous increase of the conductance between the fractional edge states at the filling factor $ν=2/5$ in comparison with the expected one for the model of independent edge states. We conclude that the model of independent fractional edge states is not suitable to describe the experimental situation at $ν=2/5$.
We study the low-lying excitations of a quantum-Hall sample that has undergone edge reconstruction such that there exist three branches of chiral edge excitations. Among the interaction processes that involve electrons close to the three Fermi points is a new type of Umklapp-scattering process which has not been discussed before. Using bosonization and a refermionization technique, we obtain exact results for electronic correlation functions and discuss the effect Umklapp scattering has on the Luttinger-liquid properties of quantum-Hall edges.
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 c…
We consider a class of interaction terms that describes correlated tunneling of composite fermions between effective Landau levels. Despite being generic and of similar strength to that of the usual density-density couplings, these terms are not included in the accepted theory of the edges of fractional quantum Hall systems. Here we show that they may lead to an instability of the edge towards a new reconstructed state with additional channels, and thereby demonstrate the incompleteness of the traditional edge theory.
We propose a surface-edge state theory for half quantized Hall conductance of surface states in topological insulators. The gap opening of a single Dirac cone for the surface states in a weak magnetic field is demonstrated. We find a new surface state resides on the surface edges and carries chiral edge current, resulting in a half-quantized Hall conductance in a four-terminal setup. We also give a physical interpretation of the half quantized conductance by showing that this state is the product of splitting of a boundary bound state of massive Dirac fermions which carries a conductance quant…
During the reconstruction of the edge of a quantum Hall liquid, Coulomb interaction energy is lowered through the change in the structure of the edge. We use theory developed earlier by one of the authors [K. Yang, Phys. Rev. Lett. 91, 036802 (2003)] to calculate the electron spectral functions of a reconstructed edge, and study the consequences of the edge reconstruction for the momentum-resolved tunneling into the edge. It is found that additional excitation modes that appear after the reconstruction produce distinct features in the energy and momentum dependence of the spectral function, wh…
Despite the rapid progresses in the field of quantum spin Hall (QSH) effect, most of the QSH systems studied up to now are based on crystalline materials. Here we propose that the QSH effect can be realized in quasicrystal lattices (QLs). We show that the electronic topology of aperiodic and amorphous insulators can be characterized by a spin Bott index $B_s$. The nontrivial QSH state in a QL is identified by a nonzero spin Bott index $B_s=1$, associated with robust edge states and quantized conductance. We also map out a topological phase diagram in which the QSH state lies in between a norma…
We argue that dynamics of gapless Fractional Quantum Hall Edge states is essentially non-linear and that it features fractionally quantized solitons propagating along the edge. Observation of solitons would be a direct evidence of fractional charges. We show that the non-linear dynamics of the Laughlin's FQH state is governed by the quantum Benjamin-Ono equation. Non-linear dynamics of gapless edge states is determined by gapped modes in the bulk of FQH liquid. The dispersion of edge modes is traced to the double boundary layer of FQH states.
The quantum Hall effect is studied in the topological insulator BiSbTeSe$_2$. By employing top- and back-gate electric fields at high magnetic field, the Landau levels of the Dirac cones in the top and bottom topological surface states can be tuned independently. When one surface is tuned to the electron-doped side of the Dirac cone and the other surface to the hole-doped side, the quantum Hall edge channels are counter-propagating. The opposite edge mode direction, combined with the opposite helicities of top and bottom surfaces, allows for scattering between these counter-propagating edge mo…
Spin and orbital freedoms of electrons traveling on spin-resolved quantum Hall edge states (quantum Hall ferromagnets) are maximally entangled. The unitary operations on these two freedoms are hence equivalent, which means one can manipulate the spins with non-magnetic methods through the orbitals. If one takes the quantization axis of spins along the magnetization axis, the zenith angle is determined by the partition rate of spin-separated edges while the azimuth angle is defined as the phase difference between the edges. Utilizing these properties, we have realized electrically controlled un…
We show evidence of the backscattering of quantum Hall edge channels in a narrow graphene Hall bar, induced by the gating effect of the conducting tip of a Scanning Gate Microscope, which we can position with nanometer precision. We show full control over the edge channels and are able, due to the spatial variation of the tip potential, to separate co-propagating edge channels in the Hall bar, creating junctions between regions of different charge carrier density, that have not been observed in devices based on top- or split-gates. The solution of the corresponding quantum scattering problem i…
