Quantum Hall Edge States Research Articles

Experimental observation of surface states and Landau levels bending in bilayer graphene

We report on microscopic measurements of the low-energy electronic structures both at zigzag and armchair edges of bilayer graphene using scanning tunneling microscopy and spectroscopy (STM and STS). We have found that, both in the absence and in the presence of a magnetic field, an almost zero-energy peak in density of states was localized at zigzag edges, as expected for the surface states at zigzag edges of bilayer graphene. In the quantum Hall regime, we have observed clearly Landau levels bending away from the charge neutrality point near both the zigzag and armchair edges. Such a result is a direct evidence for the evolution of Landau levels into the quantum Hall edge states in graphene bilayers. Our experiment indicates that it is possible to explore rich quantum Hall physics in graphene systems using STM and STS.

Long-distance entanglement of spin qubits via quantum Hall edge states

The implementation of a functional quantum computer involves entangling and coherent manipulation of a large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the most investigated solid-state qubits at present, architectural challenges are often encountered in the design of quantum circuits attempting to assemble the qubits within the very limited space available. Here, we provide a solution to such challenges based on an approach to realizing entanglement of spin qubits over long distances. We show that long-range Ruderman-Kittel-Kasuya-Yosida interaction of confined electron spins can be established by quantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and can be either Ising-type or XY-type, depending on the spin polarization of the edge state. Such a property, combined with the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gain valuable insights into the topological nature of various quantum Hall states.

Signatures of single quantum dots in graphene nanoribbons within the quantum Hall regime.

We report on the observation of periodic conductance oscillations near quantum Hall plateaus in suspended graphene nanoribbons. They are attributed to single quantum dots that are formed in the narrowest part of the ribbon, in the valleys and hills of a disorder potential. In a wide flake with two gates, a double-dot system's signature has been observed. Electrostatic confinement is enabled in single-layer graphene due to the gaps that are formed between the Landau levels, suggesting a way to create gate-defined quantum dots that can be accessed with quantum Hall edge states.

Topological Superfluid and Majorana Zero Modes in Synthetic Dimension.

Recently it has been shown that multicomponent spin-orbit-coupled fermions in one-dimensional optical lattices can be viewed as spinless fermions moving in two-dimensional synthetic lattices with synthetic magnetic flux. The quantum Hall edge states in these systems have been observed in recent experiments. In this paper we study the effect of an attractive Hubbard interaction. Since the Hubbard interaction is long-range in the synthetic dimension, it is able to efficiently induce Cooper pairing between the counterpropagating chiral edge states. The topological class of the resultant one-dimensional superfluid is determined by the parity (even/odd) of the Chern number in the two-dimensional synthetic lattice. We also show the presence of a chiral symmetry in our model, which implies Z classification and the robustness of multiple zero modes when this symmetry is unbroken.

Chiral electromagnetic waves in Weyl semimetals

We show that Weyl semimetals with broken time-reversal symmetry can host chiral electromag- netic waves. The magnetization that results in a momentum space separation of a pair of opposite chirality Weyl nodes is also responsible for the non-zero gyrotropy parameter in the system. It is then shown that a chiral electromagnetic wave can propagate in a region of space where the gyrotropy parameter changes sign. Such waves are analogs of quantum Hall edge states for photons.

Time-Dependent Modelling of Single-Electron Interferometry with Edge-States

We simulate the time-resolved dynamics of localized electrons in a 2DEG system, where an external magnetic field creates quantum Hall edge states, and properly polarized split gates define a Mach-Zehnder electron interferometer. The carriers travelling inside the Hall channels consist of localized wave packets of edge states: they are propagated numerically bymeans of a Fourier split-step approach. We find that the energy-dependent, scattering process at the quantum point contacts, together with the finite energy distribution of the carriers, have a remarkable effect on the transmission coefficient T of the device. We provide an analytical model to justify the characteristics of T which is in good agreement with the numerical simulations.

Heat diode and engine based on quantum Hall edge states

We investigate charge and energy transport in a three-terminal quantum Hall conductor. The peculiar properties of chiral propagation along the edges of the sample have important consequences on the response to thermal biases. Based on the separation of charge and heat flows, thermoelectric conversion and heat rectification can be manipulated by tuning the scattering at gate-modulated constrictions. Chiral motion in a magnetic field allows for a different behavior of left- and right-moving carriers giving rise to thermal rectification by redirecting the heat flows. We propose our system both as an efficient heat-to-work converter and as a heat diode.

Magnetization Signatures of Light-Induced Quantum Hall Edge States.

Circularly polarized light opens a gap in the Dirac spectrum of graphene and topological insulator (TI) surfaces, thereby inducing a quantum Hall-like phase. We propose to detect the accompanying edge states and their current by the magnetic field they produce. The topological nature of the edge states is reflected in the mean orbital magnetization of the sample, which shows a universal linear dependence as a function of a generalized chemical potential-independent of the driving details and the properties of the material. The proposed protocol overcomes several typically encountered problems in the realization and measurement of Floquet phases, including the destructive effects of phonons and coupled electron baths and provides a way to occupy the induced edge states selectively. We estimate practical experimental parameters and conclude that the magnetization signature of the Floquet topological phase may be detectable with current techniques.

Equilibration in a chiral Luttinger liquid

We explore the weak-strong-coupling Bose-Fermi duality in a model of a single-channel integer or fractional quantum Hall edge state with a finite-range interaction. The system is described by a chiral Luttinger liquid with nonlinear dispersion of bosonic and fermonic excitations. We use the bosonization, a unitary transformation, and a refermionization to map the system onto that of weakly interacting fermions at low temperature $T$ or weakly interacting bosons at high $T$. We calculate the equilibration rate which is found to scale with temperature as ${T}^{5}$ and ${T}^{14}$ in the high-temperature (``bosonic'') and the low-temperature (``fermonic'') regimes, respectively. The relaxation rate of a hot particle with the momentum $k$ in the fermonic regime scales as ${k}^{7}{T}^{7}$.

Chiral thermoelectrics with quantum Hall edge states.

The thermoelectric properties of a three-terminal quantum Hall conductor are investigated. We identify a contribution to the thermoelectric response that relies on the chirality of the carrier motion rather than on spatial asymmetries. The Onsager matrix becomes maximally asymmetric with configurations where either the Seebeck or the Peltier coefficients are zero while the other one remains finite. Reversing the magnetic field direction exchanges these effects, which originate from the chiral nature of the quantum Hall edge states. The possibility to generate spin-polarized currents in quantum spin Hall samples is discussed.

Optoelectronic transport through quantum Hall edge states

We use GaAs-based quantum point contacts as mesoscopic detectors to locally analyze the flow of photogenerated electrons in a two-dimensional electron gas (2DEG) at perpendicular, quantizing magnetic fields. The 2DEG is formed within a quantum well of a doped GaAs/AlGaAs-heterostructure. We find an optoelectronic signal along the lateral boundaries of the 2DEG, which is consistent with an optically induced quantum transport through quantum Hall edge channels. We demonstrate that photogenerated electrons can be directly injected into an edge channel, transported across several tens of micrometers and read-out on-chip by the quantum point contact.

Detection of spin entanglement via spin-charge separation in crossed Tomonaga-Luttinger liquids.

We investigate tunneling between two spinful Tomonaga-Luttinger liquids (TLLs) realized, e.g., as two crossed nanowires or quantum Hall edge states. When injecting into each TLL one electron of opposite spin, the dc current measured after the crossing differs for singlet, triplet, or product states. This is a striking new non-Fermi liquid feature because the (mean) current in a noninteracting beam splitter is insensitive to spin entanglement. It can be understood in terms of collective excitations subject to spin-charge separation. This behavior may offer an easier alternative to traditional entanglement detection schemes based on current noise, which we show to be suppressed by the interactions.

Photonic simulation of topological superconductor edge state and zero-energy mode at a vortex.

Photonic simulations of quantum Hall edge states and topological insulators have inspired considerable interest in recent years. Interestingly, there are theoretical predictions for another type of topological states in topological superconductors, but debates over their experimental observations still remain. Here we investigate the photonic analogue of the px + ipy model of topological superconductor. Two essential characteristics of topological superconductor, particle-hole symmetry and px + ipy pairing potentials, are well emulated in photonic systems. Its topological features are presented by chiral edge state and zero-energy mode at a vortex. This work may fertilize the study of photonic topological states, and open up the possibility for emulating wave behaviors in superconductors.

Exotic circuit elements from zero-modes in hybrid superconductor–quantum-Hall systems

Heterostructures formed by quantum Hall systems and superconductors have recently been shown to support widely coveted Majorana fermion zero-modes and still more exotic `parafermionic' generalizations. Here we establish that probing such zero-modes using quantum Hall edge states yields non-local transport signatures that pave the way towards a variety of novel circuit elements. In particular, we demonstrate quite generally that at low energies the zero-modes convert chirally moving quasiparticles into oppositely charged quasiholes propagating in the same direction---that is, they swap the sign of the chiral edge currents. One may then construct new and potentially useful circuit elements using this `perfect Andreev conversion' process, including superconducting current and voltage mirrors as well as transistors for fractional charge currents. Characterization of these circuit elements should provide striking evidence of the zero-mode physics.

Fluctuations and entanglement spectrum in quantum Hall states

The measurement of quantum entanglement in many-body systems remains challenging. One experimentally relevant fact about quantum entanglement is that in systems whose degrees of freedom map to free fermions with conserved total particle number, exact relations hold relating the full counting statistics associated with the bipartite charge fluctuations and the sequence of Rényi entropies. We draw a correspondence between the bipartite charge fluctuations and the entanglement spectrum, through the Rényi entropies. In the case of the integer quantum Hall effect, we show that it is possible to reproduce the generic features of the entanglement spectrum from a measurement of the second charge cumulant only. Additionally, asking whether it is possible to extend the free fermion result to the ν = 1/3 fractional quantum Hall case, we provide numerical evidence that the answer is negative in general. We further address the problem of quantum Hall edge states described by a Luttinger liquid, and derive expressions for the spectral functions of the real space entanglement spectrum at a quantum point contact realized in a quantum Hall sample.

Quantum Nernst engines

We theoretically propose Nernst engines based on quantum Hall edge states. We identify a setup that exhibits an extreme asymmetry between the off-diagonal Onsager coefficients for heat and charge transport. In terms of thermodynamic efficiency, this engine outperforms a recently proposed classical Nernst engine. A second setup using an antidot is found to be more efficient as energy filtering becomes less strong.

Electrically tunable spin filtering for electron tunneling between spin-resolved quantum Hall edge states and a quantum dot

Spin filtering with electrically tunable efficiency is achieved for electron tunneling between a quantum dot and spin-resolved quantum Hall edge states by locally gating the two-dimensional electron gas (2DEG) leads near the tunnel junction to the dot. The local gating can change the potential gradient in the 2DEG and consequently the edge state separation. We use this technique to electrically control the ratio of the dot–edge state tunnel coupling between opposite spins and finally increase spin filtering efficiency up to 91%, the highest ever reported, by optimizing the local gating.

Selective equilibration of spin-polarized quantum Hall edge states in graphene.

We report on transport measurements of dual-gated, single-layer graphene devices in the quantum Hall regime, allowing for independent control of the filling factors in adjoining regions. Progress in device quality allows us to study scattering between edge states when the fourfold degeneracy of the Landau level is lifted by electron correlations, causing edge states to be spin and/or valley polarized. In this new regime, we observe a dramatic departure from the equilibration seen in more disordered devices: edge states with opposite spins propagate without mixing. As a result, the degree of equilibration inferred from transport can reveal the spin polarization of the ground state at each filling factor. In particular, the first Landau level is shown to be spin polarized at half filling, providing an independent confirmation of a conclusion of Young et al. [Nat. Phys. 8, 550 (2012). The conductance in the bipolar regime is strongly suppressed, indicating that copropagating edge states, even with the same spin, do not equilibrate along PN interfaces. We attribute this behavior to the formation of an insulating ν = 0 stripe at the PN interface.

Annihilation of Colliding Bogoliubov Quasiparticles Reveals their Majorana Nature

The single-particle excitations of a superconductor are coherent superpositions of electrons and holes near the Fermi level, called Bogoliubov quasiparticles. They are Majorana fermions, meaning that pairs of quasiparticles can annihilate. We calculate the annihilation probability at a beam splitter for chiral quantum Hall edge states, obtaining a 1±cosϕ dependence on the phase difference ϕ of the superconductors from which the excitations originated (with the ± sign distinguishing singlet and triplet pairing). This provides for a nonlocal measurement of the superconducting phase in the absence of any supercurrent.

LESSONS OF NANOELECTRONICS: HALL EFFECT AND MEASUREMENT OF ELECTROCHEMICAL POTENTIALS WITHIN «BOTTOM – UP» APPROACH

Classical and quantum Hall effects, measurement of electrochemical potentials, the Landauer formulas and Buttiker formula, measurement of Hall potential, an account of magnetic field in the NEGF method, quantum Hall effect, Landau levels, and edge states in graphene are discussed in the frame of the «bottom – up» approach of modern nanoelectronics in continuation to the previous tutorial review articles of the authors.