Helical Edge Research Articles

Quantum Anomalous Hall Phase and Time‐Reversal‐Symmetry‐Broken Quantum Spin Hall Phase on the Square–Hexagon Lattice

The topological properties of the square–hexagon lattice are studied when the time‐reversal (TR) symmetry is broken. Based on the Chern number and spin Chern number, the topological trivial and nontrivial phases can be determined. When the spin‐dependent and ‐independent staggered potentials are taken into consideration, rich phase diagrams are obtained. A number of topological nontrivial phases, including the TR‐symmetry‐broken quantum spin Hall phase and quantum anomalous Hall (QAH) phase, can be achieved by tuning the spin‐dependent staggered potential. When the spin‐independent staggered potential is varied, it is possible to observe the topological phase transition from the metal phase to the QAH phase. The evolutions of the bulk bandgap and spin spectrum gap are also presented together. To support the identification of the topological phases, helical and chiral edge states are given numerically. Moreover, the corresponding density distribution and spin polarization are analyzed and discussed.

Emergent helical edge states in a hybridized three-dimensional topological insulator

As the thickness of a three-dimensional (3D) topological insulator (TI) becomes comparable to the penetration depth of surface states, quantum tunneling between surfaces turns their gapless Dirac electronic structure into a gapped spectrum. Whether the surface hybridization gap can host topological edge states is still an open question. Herein, we provide transport evidence of 2D topological states in the quantum tunneling regime of a bulk insulating 3D TI BiSbTeSe2. Different from its trivial insulating phase, this 2D topological state exhibits a finite longitudinal conductance at ~2e2/h when the Fermi level is aligned within the surface gap, indicating an emergent quantum spin Hall (QSH) state. The transition from the QSH to quantum Hall (QH) state in a transverse magnetic field further supports the existence of this distinguished 2D topological phase. In addition, we demonstrate a second route to realize the 2D topological state via surface gap-closing and topological phase transition mechanism mediated by a transverse electric field. The experimental realization of the 2D topological phase in a 3D TI enriches its phase diagram and marks an important step toward functionalized topological quantum devices.

Characterization of helical states in semiconductor quantum wells using quantum information quantities

The information content of one-electron bulk and edge states in semiconductor quantum wells is calculated in the inverted regime, where edge states, topologically protected, are responsible for the conduction in Spin Quantum Hall effect experiments. To study the information content of these states we first calculate realistic two dimensional one-electron states, solving first the eight-band $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian to obtain the bulk states and then a four band effective Hamiltonian to obtain the edge states. The behavior of information-like quantities, as a function of the different parameters that define the quantum well, is analyzed. The results presented show that the helical edge states can be singled out using different quantities that characterize the rich phenomenology of these states.

Robustness of helical edge states under edge reconstruction

The helical edge states of time-reversal invariant two-dimensional topological insulators are protected against backscattering in idealized models. In more realistic scenarios with a shallow confining potential at the sample boundary, additional strongly interacting edge states may arise, that could interfere with the topological protection of edge conduction. We find that interaction effects within the reconstructed edges are well described by the Luttinger liquid model. While interactions between this Luttinger liquid and the helical edge states can in principle give rise to dynamical spin polarization and the breaking of time-reversal symmetry, we demonstrate that random spin-orbit coupling strongly suppresses such dynamical spin polarization, resulting in the persistence of near quantized edge conduction.

Quantum Monte Carlo at the graphene quantum Hall edge

We study a continuum model of the interface of graphene and vacuum in the quantum hall regime via sign-problem-free quantum Monte Carlo, allowing us to investigate the interplay of topology and strong interactions in a graphene quantum Hall edge for large system sizes. We focus on the topological phase transition from the spin polarized state with symmetry protected gapless helical edges to the fully charge gapped canted-antiferromagnet state with spontaneous symmetry breaking, driven by the Zeeman energy. Our large system size simulations allow us to detail the behaviour of various quantities across this transition that are amenable to be probed experimentally, such as the spatially and energy-resolved local density of states and the local compressibility. We find peculiar kinks in the branches of the edge dispersion, and also an unexpected large charge susceptibility in the bulk of the canted-antiferromagnet associated with its Goldstone mode.

Topological materials for full-vector elastic waves.

Elastic wave manipulation is important in a wide variety of applications, including information processing in small elastic devices and noise control in large solid structures. The recent emergence of topological materials has opened new avenues for modulating elastic waves in solids. However, because of the full-vector feature and the complicated couplings of the longitudinal and transverse components of elastic waves, manipulating elastic waves is generally difficult compared with manipulating acoustic waves (scalar waves) and electromagnetic waves (vectorial waves but transverse only). To date, topological materials, including insulators and semimetals, have been used for acoustic and electromagnetic waves. Although topological materials with elastic waves have also been reported, the observed topological edge modes lie on the domain wall. A natural question arises: Is there an elastic metamaterial with topological edge modes on its own boundary? Here, we report a 3D metal-printed bilayer metamaterial that topologically insulates elastic waves. By introducing chiral interlayer couplings, the spin-orbit couplings for elastic waves are induced, which give rise to nontrivial topological properties. Helical edge states with vortex features were demonstrated on the boundary of the single topological phase. We further show a heterostructure of the metamaterial that exhibits tunable edge transport. Our findings could be used in devices based on elastic waves in solids.

Helical Luttinger Liquid on the Edge of a Two-Dimensional Topological Antiferromagnet

A boundary helical Luttinger liquid (HLL) with broken bulk time-reversal symmetry belongs to a unique topological class that may occur in antiferromagnets (AFM). Here, we search for signatures of HLL on the edge of a recently discovered topological AFM, MnBi2Te4 even-layer. Using a scanning superconducting quantum interference device, we directly image helical edge current in the AFM ground state appearing at its charge neutral point. Such a helical edge state accompanies an insulating bulk which is topologically distinct from the ferromagnetic Chern insulator phase, as revealed in a magnetic field driven quantum phase transition. The edge conductance of the AFM order follows a power law as a function of temperature and source-drain bias which serves as strong evidence for HLL. Such HLL scaling is robust at finite fields below the quantum critical point. The observed HLL in a layered AFM semiconductor represents a highly tunable topological matter compatible with future spintronics and quantum computation.

Two dimensional LiMgAs: A topological quantum catalyst for hydrogen evolution reaction

Quantum materials, such as topological insulators (TIs), are promising due to diverse applications of their robust surface/edge states in the bulk three-dimensional (3D) and two-dimensional (2D) regimes. Such conducting surface states in 3D systems host “electron baths,” which are known to facilitate catalysis. However, the analogous effects in 2D scenarios wherein conducting helical edge states leading to Fermionic accumulation have been scarcely addressed. Using first-principles calculations, we demonstrate that the conducting edge states in 2D TIs, such as LiMgAs, can be exploited to facilitate excellent catalytic response toward hydrogen evolution reactions. The Gibbs free energy in such cases was found to be as low as −0.02 eV, which is quite superior compared to other materials reported in the literature. The concept presented herein can be extended to other well-known 2D TIs and used to realize unconventional topological quantum catalysts for ultra-high performance and efficient catalytic applications.

Effect of magnetic impurity scattering on transport in topological insulators

Charge transport in topological insulators is primarily characterised by so-called topologically projected helical edge states, where charge carriers are correlated in spin and momentum. In principle, dissipation-less current can be carried by these edge states as backscattering from impurities and defects is suppressed as long as time-reversal symmetry is not broken. However, applied magnetic fields or underlying nuclear spin-defects in the substrate can break this time reversal symmetry. In particular, magnetic impurities lead to back-scattering by spin-flip processes. We have investigated the effects of point-wise magnetic impurities on the transport properties of helical edge states in the BHZ model using the Non-Equilibrium Green's Function formalism and compared the results to a semi-analytic approach. Using these techniques we study the influence of impurity strength and spin impurity polarization. We observe a secondary effect of defect-defect interaction that depends on the underlying material parameters which introduces a non-monotonic response of the conductance to defect density. This in turn suggests a qualitative difference in magneto-transport signatures in the dilute and high density spin impurity limits.

Pseudospin topological phase transition induced by rotation operation in two-dimensional dielectric photonic crystal with C6 symmetry

We construct a kind of two-dimensional photonic crystal with triangular lattice in which the primitive cell contains six identical rods composed of two half cylinders with different dielectric materials. All the rods have the same rotations around themselves, ensuring that the system preserves the C6 symmetry. The pseudospin topological phase transition (TPT) of the system is realized by rotating the rod elements, and the helical edge states (HESs) are constructed along the interface between two photonic crystals with opposite rotations. Numerical results and simulations reveal that the HESs are topologically protected with the pseudospin-locked unidirectional propagation of electromagnetic waves with robustness against corners and defects along the interface. Adjusting the size of rod elements, the TPT can be engineered at needed frequency.

Geometric Modeling of Serrated Cutters for Endmilling Forces using Local Oblique Cutting and Rake Normal Correspondence

For nonstandard endmills with sinusoidal and circular serrations on nominally helical cutting edges, the edge space curve was subdivided into straight edge elements for constant increments of arc length. To calculate the local normal rake angle at the serrated edge point, Local Rake Normal Correspondence used the normal vector on the rake face of the corresponding unserrated helical edge point at the same axial cross section. Uncut chip thickness h was computed under Local Oblique Cutting Correspondence published earlier. Stabler’s rule specified chip flow direction and Lazoglu mechanistic normal and friction force coefficients completed the straight edge cutting element forces. Local force contribution vectors summed over the tool engagement resulted in global tool forces. Number of straight edge element computations were minimized with the hypotheses (a) cutting predominantly occurs at the vicinity of serration tips and (b) only the stock surfaces swept by flutes immediately prior and following the current flute need be considered for h. Interaction of axial cross sections at a significant fraction of the serration wavelength was noted. Thus, treating tool axial subdivisions in relative isolation for h is untenable. However, local normal rake angle on the serrated edges followed earlier published trends. For a square endmill, sinusoidally serrated, cutting with low axial depth of cut ADOC, modeled forces showed idle periods and unbalance fluctuations during a full rotation, but not the data; modeled forces for an identical unserrated helical cutter were lower than the serrated. For the same endmill with circular serrations cutting at high ADOC, no idle periods or unbalance were predicted and the agreement with published data was better.

Gate-mediated transition between antiferromagnetic topological and Chern insulators in honeycomb X3MnN3 (X=Sr,Ba)

Interest in antiferromagnetic (AFM) spintronics has been greatly stimulated by both the complex interplay between antiferromagnetism and band topology and the electrical switching of exotic properties they give rise to. However, the topological phase transitions are usually associated with magnetic transitions where the AFM ordering may be heavily deformed and limit their promising applications. Here, by first-principles calculations and a tight-binding model, we show that an electrically controlled topological phase transition with robust AFM ordering is possible in a honeycomb lattice, and remarkably the long-awaited AFM quantum anomalous Hall effect is obtained. The calculated spin Chern number and Chern number for a two-dimensional AFM topological insulator and AFM Chern insulator are, respectively, ${\mathcal{C}}_{\mathcal{S}}=1$ and $\mathcal{C}=1$, which are further explicitly confirmed by the corresponding helical and chiral edge states. Moreover, we identify intrinsic AFM ${X}_{3}\mathrm{Mn}{\mathrm{N}}_{3}$ $(X=\mathrm{Sr},\mathrm{Ba})$ bilayers as probable candidates for the experimental realization of the proposed phenomenon, and unexpectedly point out that the valley polarization of AFM ${X}_{3}\mathrm{Mn}{\mathrm{N}}_{3}$ is much larger than that of the known ferrovalley systems, indicating the high possibility of innovative applications in AFM spintronics and valleytronics.

Corner- and sublattice-sensitive Majorana zero modes on the kagome lattice

In a first-order topological phase with sublattice degrees of freedom, a change in the boundary sublattice termination has no effect on the existence of gapless boundary states in dimensions higher than one. However, such a change may strongly affect the physical properties of those boundary states. Motivated by this observation, we perform a systematic study of the impact of sublattice terminations on the boundary physics on the two-dimensional kagome lattice. We find that the energies of the Dirac points of helical edge states in two-dimensional first-order topological kagome insulators sensitively depend on the terminating sublattices at the edge. Remarkably, this property admits the realization of a time-reversal invariant second-order topological superconducting phase with highly controllable Majorana Kramers pairs at the corners and sublattice domain walls by putting the topological kagome insulator in proximity to a $d$-wave superconductor. Moreover, substituting the $d$-wave superconductor with a conventional $s$-wave superconductor, we find that highly controllable Majorana zero modes can also be realized at the corners and sublattice domain walls if an in-plane Zeeman field is additionally applied. Our study reveals promising platforms to implement highly controllable Majorana zero modes.

Sub- to super-Poissonian crossover of current noise in helical edge states coupled to a spin impurity in a magnetic field

Edge states of two-dimensional topological insulators are helical and single-particle backscattering is prohibited by time-reversal symmetry. In this paper, we show that an isotropic exchange coupling of helical edge states (HES) to a spin 1/2 impurity subjected to a magnetic field results in characteristic backscattering current noise (BCN) as a function of bias voltage and tilt angle between the direction of the magnetic field and the quantization axis of the HES. In particular, we find transitions from sub-Poissonian (antibunching) to super-Poissonian (bunching) behavior as a direct consequence of the helicity of the edge state electrons. We use the method of full counting statistics within a master equation approach treating the exchange coupling between the spin-1/2 impurity and the HES perturbatively. We express the BCN via coincidence correlation functions of scattering processes between the HES, which gives a precise interpretation of the Fano factor in terms of bunching and antibunching behavior of electron jump events. We also investigate the effect of electron-electron interactions in the HES in terms of the Tomonaga-Luttinger liquid theory.

Comparison and evaluation of microchannel array machining with structured cutter

ABSTRACT The wide application of microchannel arrays promotes the development of their fabrication in the direction of simplicity, low cost, and high quality. In this paper, a method of machining parallel microchannels with equal width and spacing by a cutter with multiparallel helical edge is proposed, and the response of the cutter when machining materials with different hardness (copper and nickel-based superalloy Inconel718) is explored. The tool takes only 4.5 minutes to manufacture, a significant cost saving. The time-domain signal characteristics, surface quality, burr characteristics, dimensional accuracy, and tool wear state of milling different materials are compared and analyzed. In addition, the characteristics of the chip produced by the tool are discussed. The results show that parallel microchannels with a dimensional accuracy of 118.7 ± 1.35 μm and 118.7 ± 3.61 μm can be fabricated on In718 and copper, respectively. This research is expected to be used to fabricate parallel microchannels in a variety of materials and has important application value to promote the efficient and low-cost processing of small-sized microchannel arrays.

Chiral detection of Majorana bound states at the edge of a quantum spin Hall insulator

A hybrid setup consisting of a superconductivity-proximitized quantum spin Hall insulator and a quantum anomalous Hall (QAH) insulator is proposed for chiral injection of electrons into the Majorana bound state (MBS). An unexplored region of the phase space involving the Zeeman-field-induced boost of the helical edge state is then proposed for the detection of the MBS. Two-dimensional transport simulations of our proposed setup are compared with the corresponding setup in the absence of the QAH region, when moderate potential and magnetic disorder are included. The remarkable contrast between the two results demonstrates the possibility for an unprecedented immunity from disorder-induced masking of the MBS detection in our proposed setup.

Two-dimensional anisotropic Dirac materials PtN4C2 and Pt2N8C6 with quantum spin and valley Hall effects

We propose two different two-dimensional topological Dirac materials, planar ${\mathrm{PtN}}_{4}{\mathrm{C}}_{2}$ and ${\mathrm{Pt}}_{2}{\mathrm{N}}_{8}{\mathrm{C}}_{6}$, which exhibit graphenelike electronic structures with linearly dispersive Dirac-cone states exactly at the Fermi level. Moreover, the Dirac cone is anisotropic, resulting in anisotropic Fermi velocities and making it possible to realize orientation-dependent quantum devices. Using first-principles electronic structure calculations, we have systemically studied the structural, electronic, and topological properties. We find that spin-orbit coupling opens a sizable topological band gap so that the materials can be classified as quantum spin Hall insulators as well as quantum valley Hall insulators. Helical edge states that reside in the insulating band gap connecting the bulk conduction and valence bands are observed. Our work not only expands the Dirac-cone material family, but also provides another avenue to search for more two-dimensional topological quantum spin and valley Hall insulators.

Evidence of a room-temperature quantum spin Hall edge state in a higher-order topological insulator

Room-temperature realization of macroscopic quantum phases is one of the major pursuits in fundamental physics1,2. The quantum spin Hall phase3-6 is a topological quantum phase that features a two-dimensional insulating bulk and a helical edge state. Here we use vector magnetic field and variable temperature based scanning tunnelling microscopy to provide micro-spectroscopic evidence for a room-temperature quantum spin Hall edge state on the surface of the higher-order topological insulator Bi4Br4. We find that the atomically resolved lattice exhibits a large insulating gap of over 200 meV, and an atomically sharp monolayer step edge hosts an in-gap gapless state, suggesting topological bulk-boundary correspondence. An external magnetic field can gap the edge state, consistent with the time-reversal symmetry protection inherent in the underlying band topology. We further identify the geometrical hybridization of such edge states, which not only supports the Z2 topology of the quantum spin Hall state but also visualizes the building blocks of the higher-order topological insulator phase. Our results further encourage the exploration of high-temperature transport quantization of the putative topological phase reported here.

Linear and nonlinear thermoelectric transport in a quantum spin Hall insulators coupled with a nanomagnet

Thermoelectric effects in quantum systems have been focused in recent years. Thermoelectric energy conversion study of systems with edge states, such as quantum Hall insulators and quantum spin Hall insulators, is one of the most important frontier topics in material science and condensed-matter physics. Based on the previous paper (Gresta in Phys Rev Lett 123:186801, 2019), we further investigated the linear and nonlinear thermoelectric transport properties of helical edge states of the quantum spin Hall insulators coupled with double nanomagnet, calculated the Seebeck coefficients S_c and the thermoelectrical figure of merit ZT, discussed the influence of the length of the nanomagnet and the relative tilt angle of component of the magnetization perpendicular on the thermoelectric coefficients (S_c and ZT), and summarized some meaningful conclusions in the linear response regime. In the nonlinear regime, we calculated the equivalent figure of merit ZT_M and the power-generation efficiency eta in different length of the nanomagnet, obtain the temperature difference of achieving optimal thermoelectricity. The results of this paper further confirm that the setup can indeed be used as a device for achieving high performance thermoelectric.

Interplay of Leakage Radiation and Protection in Topological Photonic Crystal Cavities

AbstractThe introduction of topological concepts to the design of photonic crystal cavities holds great promise for applications in integrated photonics due to the prospect of topological protection. This study examines the signatures of topological light confinement in the leakage radiation of 2D topological photonic crystal cavities. The cavities are implemented in an all‐dielectric platform that features the photonic quantum spin Hall effect at telecom wavelengths and supports helical edge states that are weakly coupled to the radiation continuum. The modes of resonators scaling down to single point defects in the surrounding bulk lattice are characterized via spectral position and multipolar nature of the eigenstates. The mode profiles in real and momentum space are mapped using far‐field imaging and Fourier‐spectropolarimetry, revealing how certain properties of the cavity modes reflect on their origin in the topological bandstructure. This includes band‐inversion‐induced confinement and inverted scaling of mode spectra for trivial and topological defect cavities. Furthermore, hallmarks of topological protection in the loss rates are demonstrated, which are largely unaffected by cavity shape and size. The results constitute an important step toward the use of radiative topological cavities for on‐chip confinement of light, control of emitted wave fronts, and enhancement of light–matter interactions.