Wave Functions Of Edge States Research Articles

Dynamics of the Entanglement Zero Modes in the Haldane Model under a Quantum Quench

We investigate evolution of entanglement spectra of the Haldane model for Chern insulators upon a sudden quench within the same topological phase. In particular, we focus on the location of the entanglement spectrum crossing, which signifies the bulk topology. It is shown that the coplanarity condition for the pseudomagnetic field of the model, which can be used to determine the crossing in the equilibrium case, needs to be relaxed. We analytically derive the non-equilibrium condition with the help of an edge-state wave function ansatz and a dynamically induced length-scale cutoff. With spectral analyses, it is realized that the oscillatory behavior of the crossing is dominated by the interband excitations at the van Hove singularities.

Controllable quantum point junction on the surface of an antiferromagnetic topological insulator

Engineering and manipulation of unidirectional channels has been achieved in quantum Hall systems, leading to the construction of electron interferometers and proposals for low-power electronics and quantum information science applications. However, to fully control the mixing and interference of edge-state wave functions, one needs stable and tunable junctions. Encouraged by recent material candidates, here we propose to achieve this using an antiferromagnetic topological insulator that supports two distinct types of gapless unidirectional channels, one from antiferromagnetic domain walls and the other from single-height steps. Their distinct geometric nature allows them to intersect robustly to form quantum point junctions, which then enables their control by magnetic and electrostatic local probes. We show how the existence of stable and tunable junctions, the intrinsic magnetism and the potential for higher-temperature performance make antiferromagnetic topological insulators a promising platform for electron quantum optics and microelectronic applications.

Edge-State Wave Functions from Momentum-Conserving Tunneling Spectroscopy.

We perform momentum-conserving tunneling spectroscopy using a GaAs cleaved-edge overgrowth quantum wire to investigate adjacent quantum Hall edge states. We use the lowest five wire modes with their distinct wave functions to probe each edge state and apply magnetic fields to modify the wave functions and their overlap. This reveals an intricate and rich tunneling conductance fan structure which is succinctly different for each of the wire modes. We self-consistently solve the Poisson-Schrödinger equations to simulate the spectroscopy, reproducing the striking fans in great detail, thus, confirming the calculations. Further, the model predicts hybridization between wire states and Landau levels, which is also confirmed experimentally. This establishes momentum-conserving tunneling spectroscopy as a powerful technique to probe edge state wave functions.

Effects of Electron-Hole Asymmetry on Electronic Structure of Helical Edge States in HgTe/HgCdTe Quantum Wells

We study the effects of electron-hole asymmetry on the electronic structure of helical edge states in HgTe/HgCdTe quantum wells. In the framework of the four-band kp-model, which takes into account the absence of a spatial inversion centre, we obtain analytical expressions for the energy spectrum and wave functions of edge states, as well as the effective g-factor tensor and matrix elements of electro-dipole optical transitions between the spin branches of the edge electrons. We show that when two conditions are simultaneously satisfied -- electron-hole asymmetry and the absence of an inversion centre -- the spectrum of edge electrons deviates from the linear one, in that case we obtain corrections to the linear spectrum.

The edge states properties of the topological insulator with fermion path nonanalyticity points located at the edge

The edge states properties of a finite-size 2D topological insulator (TI) with the open boundary conditions in both directions are studied. It is shown that fermion path nonanalyticity points appearing on the edge of finite-size TI cause a nonuniform edge state distribution along the TI edge. The character of this distribution depends substantially on the edge state’s energy position in the bulk band gap. The edge state with the energy in the middle of the gap tends to be located in the corners between two nonparallel edges of the system, while the edge states with the energy close to the bulk band edge avoid the corners. The monotonic transition from one space distribution of the edge state wave function to another along the edge state’s band is observed. The mentioned edge state’s structure leads to the nonlinear current-voltage characteristic of the square-shaped TI with contacts connected with the corners of the device. It opens the possibility to control the current with the gate voltage and is useful for construction the nano-size devices. For the edge states with energy in the middle of a bulk gap, the vortex structure of the probability flux located near corners is found.

Влияние электрон-дырочной асимметрии на электронную структуру спиральных краевых состояний в квантовой яме HgTe/HgCdTe

We study the effects of electron-hole asymmetry on the electronic structure of helical edge states in HgTe/HgCdTe quantum wells. In the framework of the four-band kp-model, which takes into account the absence of a spacial inversion centre, we obtain analytical expressions for the energy spectrum and wave functions of edge states, as well as the effective g-factor tensor and matrix elements of electro-dipole optical transitions between the spin branches of the edge electrons. We show that when two conditions - electron-hole asymmetry and the absence of an inversion centre - are simultaneously satisfied, the spectrum of edge electrons deviates from the linear one, and we obtain the corresponding corrections.

Extended SSH Model: Non-Local Couplings and Non-Monotonous Edge States

We construct a generalized system by introducing an additional long-range hopping to the well-known Su-Schrieffer-Heeger (SSH) model. This system exhibits richer topological properties including non-trivial topological phases and associated localized edge states. We study the zero-energy edge states in detail and derive the edge-state wave functions using two different methods. Furthermore, we propose a possible setup using octupole moments optically excited on an array of dielectric particles for the realization of the system, and by adjusting the coupling strengths between neighboring particles we can control the hotspots (near-field enhancement) in such structures.

Optimization of edge state velocity in the integer quantum Hall regime

Observation of interference in the quantum Hall regime may be hampered by a small edge state velocity due to finite phase coherence time. Therefore designing two quantum point contact (QPCs) interferometers having a high edge state velocity is desirable. Here we present a new simulation method for designing heterostructures with high edge state velocity by realistically modeling edge states near QPCs in the integer quantum Hall effect (IQHE) regime. Using this simulation method, we also predict the filling factor at the center of QPCs and their conductance at different gate voltages. The 3D Schr\"odinger equation is split into 1D and 2D parts. Quasi-1D Schr\"odinger and Poisson equations are solved self-consistently in the IQHE regime to obtain the potential profile, and quantum transport is used to solve for the edge state wave functions. The velocity of edge states is found to be $\ensuremath{\langle}E\ensuremath{\rangle}/B$, where $\ensuremath{\langle}E\ensuremath{\rangle}$ is the expectation value of the electric field for the edge state. Anisotropically etched trench gated heterostructures with double-sided delta doping have the highest edge state velocity among the structures considered.

Topological phase transitions in square-octagon lattice with Rashba spin-orbit coupling

Motivated by the square-octagon lattice which supports topological phases over a wide range of parameters and a number of interesting quantum phase transitions in the phase diagram when considering the intrinsic spin-orbit coupling, we investigate the topological phase transitions in the isotropic square-octagon lattice combining the effects of both spin-orbit couplings and exchange field. The inversion symmetry and time-reversal symmetry are broken when both Rashba spin-orbit coupling and exchange field are present. The <i>Z</i><sub>2</sub> index is not applicable for quantum spin Hall systems without time-reversal symmetry, but the spin Chern number remains valid even in the absence of time-reversal symmetry. Therefore, we use the Chern number and spin Chern number to describe the topological properties of the system. We explore that a variety of topologically nontrivial states appear with changing the exchange field, including time-reversal-symmetry-broken quantum spin Hall states and quantum anomalous Hall states. The phase transition between these topological phases is accompanied by the closing of band gaps. Interestingly, the quantum spin Hall effect described by nonzero spin Chern number is found to remain intact when the time-reversal symmetry is broken. Furthermore, the variation of the amplitude of the exchange field and filling factor drive interesting topological phase transitions from the time-reversal-symmetry-broken quantum spin Hall phase to spin-filtered quantum anomalous Hall phase. A spin-filtered quantum anomalous Hall phase is characterized by the presence of edge states with only one spin component, which provides an interesting route towards quantum spin manipulation. We also present the band structures, edge state wave functions, and spin polarizations of the different topological phases in the system. It is demonstrated that the energy spectra of edge states are in good agreement with the topological characterization based on the Chern number and spin Chern number. In particular, we observe that gapless edge states can appear in a time-reversal-symmetry-broken quantum spin Hall system, but the corresponding spin spectrum gap remains open on the edges. Recently, an important functional material ZnO with quasi square-octagon lattice has been found experimentally. Consequently, the results found in our work are helpful for understanding the property of square-octagon lattice and studying the real materials with square-octagon structure.

Boundary conditions of Weyl semimetals

We find that generic boundary conditions of Weyl semimetal is dictated by only a single real parameter, in the continuum limit. We determine how the energy dispersions (the Fermi arcs) and the wave functions of edge states depend on this parameter. Lattice models are found to be consistent with our generic observation. Furthermore, the enhanced parameter space of the boundary condition is shown to support a novel topological number.

Complex band structure of topologically protected edge states

One of the great successes of modern condensed matter physics is the discovery of topological insulators (TIs). A thorough investigation of their properties could bring such materials from fundamental research to potential applications. Here, we report on theoretical investigations of the complex band structure (CBS) of two-dimensional (2D) TIs. We utilize the tight-binding form of the Bernevig, Hughes, and Zhang model as a prototype for a generic 2D TI. Based on this model, we outline the conditions that the CBS must satisfy in order to guarantee the presence of topologically protected edge states. Furthermore, we use the Green's function technique to show how these edge states are localized, highlighting the fact that the decay of the edge-state wave functions into the bulk of a TI is not necessarily monotonic and, in fact, can exhibit an oscillatory behavior that is consistent with the predicted CBS of the bulk TI. These results may have implications for electronic and spin transport across a TI when it is used as a tunnel barrier.

Signatures of Rashba spin-orbit interaction in the superconducting proximity effect in helical Luttinger liquids

We consider the superconducting proximity effect in a helical Luttinger liquid at the edge of a two-dimensional (2D) topological insulator, and derive the low-energy Hamiltonian for an edge state tunnel-coupled to an $s$-wave superconductor. In addition to correlations between the left- and right-moving modes, the coupling can induce them inside a single mode, as the spin axis of the edge modes is not necessarily constant. This can be induced controllably in HgTe/CdTe quantum wells via the Rashba spin-orbit coupling, and is a consequence of the 2D nature of the edge-state wave function. The distinction of these two features in the proximity effect is also vital for the use of such helical modes in order to split Cooper pairs. We discuss the consequent transport signatures, and point out a long-ranged feature in a dc conductance measurement that can be used to distinguish the two types of correlations present and to determine the magnitude of the Rashba interaction.

High Chern number quantum anomalous Hall phases in single-layer graphene with Haldane orbital coupling

We investigate possible phase transitions among the different quantum anomalous Hall (QAH) phases in a zigzag graphene ribbon under the influence of the exchange field. The effective tight-binding Hamiltonian for graphene is made up of the hopping term, the Kane-Mele and Rashba spin-orbit couplings as well as the Haldane orbital term. We find that the variation of the exchange field results in bulk gap-closing phenomena and phase transitions occur in the graphene system. If the Haldane orbital coupling is absent, the phase transition between the chiral (anti-chiral) edge state $\nu=+2$ ($\nu=-2$) and the pseudo-quantum spin Hall state ($\nu=0$) takes place. Surprisingly, when the Haldane orbital coupling is taken into account, an intermediate QSH phase with two additional edge modes appears in between phases $\nu=+2$ and $\nu=-2$. This intermediate phase is therefore either the hyper-chiral edge state of high Chern number $\nu=+4$ or anti-hyper-chiral edge state of $\nu=-4$ when the direction of exchange field is reversed. We present the band structures, edge state wave functions and current distributions of the different QAH phases in the system. We also report the critical exchange field values for the QAH phase transitions.

Switching Spin and Charge between Edge States in Topological Insulator Constrictions

We show how the coupling between opposite edge states, which overlap in a constriction made of the topological insulator mercury telluride (HgTe), can be employed both for steering the charge flow into different edge modes and for controlled spin switching. Unlike in a conventional spin transistor, the switching does not rely on a tunable Rashba spin-orbit interaction, but on the energy dependence of the edge state wave functions. Based on this mechanism, and supported by extensive numerical transport calculations, we present two different ways to control spin and charge currents, depending on the local gating of the constriction, resulting in a high fidelity spin transistor.

Scanning microscopy of nuclear spin polarization via quantum Hall edge channels

Nuclear spin polarization is probed via hyperfine interaction with spin-split edge channels. By scanning the edge channels with a biased metal side gate, the local profile of nuclear spin polarization is imaged with a resolution of $\ensuremath{\sim}14\text{ }\text{nm}$, determined by the extension of the edge-state wave function. Distribution of the nuclear spins dynamically polarized by the edge channels is found to be localized in a region as narrow as 35 nm at the lowest current (0.1 nA) but broadens with increasing current. The decay process is studied in detail with the same technique and the temporal dynamics of the nuclear polarization are interpreted in terms of the interplay of diffusion, electron-nuclear spin interaction, and spin-lattice relaxation.

Unscreened Coulomb Interactions and the Quantum Spin Hall Phase in Neutral Zigzag Graphene Ribbons

A study of the effect of unscreened Coulomb interactions on the quantum spin Hall phase of finite-width neutral zigzag graphene ribbons is presented. By solving a tight-binding Hamiltonian that includes the intrinsic spin-orbit (ISO) interaction, exact expressions for band structures and edge-state wave functions are obtained. These analytic results, supported by tight-binding calculations, show that chiral spin-filtered edge states are composed of localized and damped oscillatory wave functions, reminiscent of the ones obtained in armchair ribbons. The addition of long-range Coulomb interactions opens a gap in the charge sector with a gapless spin sector. In contrast to armchair terminations, the charge gap vanishes exponentially with the ribbon width and its amplitude and decay length are strongly dependent on the ISO coupling. Comparison with reported ab initio calculations are presented.

Self-consistent edge states of quantum wires in high magnetic fields

Results of detailed self-consistent calculations for edge states are reported on the energy level structure of wide quantum wires formed by a split gate at GaAs/AlGaAs heterostructures in high magnetic fields at low temperatures. The flattening of the dispersion of edge states (a compressible strip) appears only in very wide wires under limited conditions that the estimated width of the strip is larger than the extent of the edge-state wave function, i.e. the cyclotron radius.

Electron structure near abrupt edges in 2DEG under strong magnetic field

AbstractEnergies and wave functions of edge states in a two‐dimensional electron gas are evaluated for a finite step potential barrier model. The spectrum, instead of smooth bending of Landau branches in the vicinity of the barrier acquires a step‐like form; unexpected edge plateaus and significant energy gap reduction take place between the neighbouring Landau branches above the barrier tops. The origin of these phenomena is traced. Stability with respect to modifications is established. Manifestation of the qualitatively new features of electron densities of states in abrupt confinements through magnetooptical and nuclear magnetic spin relaxation effects is proposed.

Edge states in a strong magnetic field

Energies and wave functions of edge states in twodimensional electron gas are evaluated for a finite step potential barrier model. The spectrum exhibits richer structure than observed previously for an infinite barrier model. Surprisingly, instead of smooth Landau level bending in the vicinity of the barrier, the levels acquire a steplike form. These plateaus have a direct impact on widths of the magnetotransport conducting channels.

Quenching of scattering in mesoscopic systems in the quantum Hall regime.

We study theoretically the elastic and inelastic scattering of the edge states of quasi-one-dimensional and quasi-two-dimensional electron systems in the presence of magnetic fields. We obtain a large mean free path (l\ifmmode\bar\else\textasciimacron\fi{}) as it has been experimentally observed. The Born approximation in the case of impurities and a deformation-potential scheme in the case of acoustic phonons is used to calculate the lifetime and mean free path of a carrier in one of the edge states. The magnitude that characterizes the efficiency of the scattering is the center of gravity of the edge-state wave function, which differs from the semiclassical center of the orbit in the ratio of the velocity of the state to the cyclotron frequency. When both inter- and intra-edge-state mechanisms for scattering coexist, the latter dominates for high fields, giving a monotonically decreasing mean free path as B increases. However, when the filling conditions just allow the inter-edge-state mechanism, l\ifmmode\bar\else\textasciimacron\fi{} has a minimum as a function of B. In this case, the elastic impurity contribution dominates and the scattering is suppressed by the magnetic field faster than exponentially. Such a quenching is much more efficient when the energy of the initial edge state is close to that of a bulk Landau level, in agreement with the available experimental information.