Quantum transport signatures of non-trivial topological edge states in a ring-shaped Su–Schrieffer–Heeger double-chain system

In the ring-shaped Su–Schrieffer–Heeger(SSH)double-chain,the quantum interference between the two different electron tunneling paths of the upper and lower chains has an important influence on the electron transport properties of non-trivial topological edge states.Here,we have studied the electron transport signatures of non-trivial topological edge states in a ring-shaped SSH double-chain system based on the wave-guide theory and transfer-matrix method.In the ringshaped SSH double-chain with the upper chain being different from the lower one,it is demonstrated that the electron transmission probability displays the four and two resonance peaks associated with the non-trivial topological edge states in the weak and strong coupling regimes,respectively.Whereas in the case of the upper chain being the same as the lower one,the two transmission resonance peaks associated with the non-trivial topological edge states in the weak coupling regime are only found,and that in the strong coupling regime disappear that originated from the destructive interference between the two different electron tunneling paths of the upper and lower chains.Consequently,the variation of the number of transmission resonance peaks associated with the non-trivial topological edge states in the weak and strong coupling regimes suggests that an alternative scheme for detecting non-trivial topological edge states in the ring-shaped SSH doublechain system.     

Spin wave vortex from the scattering on Bloch point solitons

The interaction of a spin wave with a stationary Bloch point is studied. The topological non-trivial structure of the Bloch point manifests in the propagation of spin waves endowing them with a gauge potential that resembles the one associated with the interaction of a magnetic monopole and an electron. By pursuing this analogy, we are led to the conclusion that the scattering of spin waves and Bloch points is accompanied by the creation of a magnon vortex. Interference between such a vortex and a plane wave leads to dislocations in the interference pattern that can be measurable by means of magnon holography.     

A bilayer Double Semion model with symmetry-enriched topological order

We construct a new model of two-dimensional quantum spin systems that combines intrinsic topological orders and a global symmetry called flavour symmetry. It is referred as the bilayer Doubled Semion model (bDS) and is an instance of symmetry-enriched topological order. A honeycomb bilayer lattice is introduced to combine a Double Semion Topological Order with a global spin–flavour symmetry to get the fractionalization of its quasiparticles. The bDS model exhibits non-trivial braiding self-statistics of excitations and its dual model constitutes a Symmetry-Protected Topological Order with novel edge states. This dual model gives rise to a bilayer Non-Trivial Paramagnet that is invariant under the flavour symmetry and the well-known spin flip symmetry.     

The family of topological Hall effects for electrons in skyrmion crystals

Hall effects of electrons can be produced by an external magnetic field, spin–orbit coupling or a topologically non-trivial spin texture. The topological Hall effect (THE) – caused by the latter – is commonly observed in magnetic skyrmion crystals. Here, we show analogies of the THE to the conventional Hall effect (HE), the anomalous Hall effect (AHE), and the spin Hall effect (SHE). In the limit of strong coupling between conduction electron spins and the local magnetic texture the THE can be described by means of a fictitious, “emergent” magnetic field. In this sense the THE can be mapped onto the HE caused by an external magnetic field. Due to complete alignment of electron spin and magnetic texture, the transverse charge conductivity is linked to a transverse spin conductivity. They are disconnected for weak coupling of electron spin and magnetic texture; the THE is then related to the AHE. The topological equivalent to the SHE can be found in antiferromagnetic skyrmion crystals. We substantiate our claims by calculations of the edge states for a finite sample. These states reveal in which situation the topological analogue to a quantized HE, quantized AHE, and quantized SHE can be found.     

Spin-filtered edge states in graphene

Spin–orbit coupling changes graphene, in principle, into a two-dimensional topological insulator, also known as quantum spin Hall insulator. One of the expected consequences is the existence of spin-filtered edge states that carry dissipationless spin currents and undergo no backscattering in the presence of non-magnetic disorder, leading to quantization of conductance. Whereas, due to the small size of spin–orbit coupling in graphene, the experimental observation of these remarkable predictions is unlikely, the theoretical understanding of these spin-filtered states is shedding light on the electronic properties of edge states in other two-dimensional quantum spin Hall insulators. Here we review the effect of a variety of perturbations, like curvature, disorder, edge reconstruction, edge crystallographic orientation, and Coulomb interactions on the electronic properties of these spin filtered states.     

Spin effects in edge transport in two-dimensional topological insulators

Investigations of topological insulators, which are two- and three-dimensional systems with a gap in the bulk spectrum and topologically protected gapless edge states, are of considerable fundamental interest at present. The experiments confirming the presence of the edge states in two-dimensional systems with inverted bands and problems of determining the nature of such states in these experiments are reviewed. Special attention is focused on spin-sensitive experiments since the topological edge states have a nontrivial spin structure.     

Spin polarized quantum pump effect in zigzag graphene nanoribbons

The spin polarized adiabatic quantum pump effect in zigzag graphene nanoribbons has been numerically analyzed. Since the ground state of such a ribbon is antiferromagnetic (the opposite spin electrons are located on the opposite edges of the ribbon), the spin currents can be generated in this system with the help of the quantum pump effect when symmetry between the opposite spin states is broken. Two methods of this breaking by means of defects at the ribbon edge and the transverse electric field have been proposed. It has been shown that the generation of not only the electron and spin currents, but also the purely spin current is possible in both cases.     

Spin injection from topological insulator tunnel-coupled to metallic leads

We study theoretically helical edge states of 2D and 3D topological insulators (TI) tunnel-coupled to metal leads and show that their transport properties are strongly affected by contacts as the latter play a role of a heat bath and induce damping and relaxation of electrons in the helical states of TI. A simple structure that produces a pure spin current in the external circuit is proposed. The current and spin current delivered to the external circuit depend on relation between characteristic lengths: damping length due to tunneling, contact length and, in case of 3D TI, mean free path and spin relaxation length caused by momentum scattering. If the damping length due to tunneling is the smallest one, then the electric and spin currents are proportional to the conductance quantum in 2D TI, and to the conductance quantum multiplied by the ratio of the contact width to the Fermi wavelength in 3D TI.     

Chirality in photonic systems

The optical modes of photonic structures are the so-called TE and TM modes that bring intrinsic spin–orbit coupling and chirality to these systems. This, combined with the unique flexibility of design of the photonic potential, and the possibility to mix photon states with excitonic resonances, sensitive to magnetic field and interactions, allows us to achieve many phenomena, often analogous to other solid-state systems. In this contribution, we review in a qualitative and comprehensive way several of these realizations, namely the optical spin Hall effect, the creation of spin currents protected by a non-trivial geometry, the Berry curvature for photons, and the photonic/polaritonic topological insulator.     

The topological quantum phase transitions in Lieb lattice driven by the Rashba SOC and exchange field

The quantum spin Hall (QSH) effect and the quantum anomalous Hall (QAH) effect in Lieblattice are investigated in the presence of both Rashba spin-orbit coupling (SOC) anduniform exchange field. The Lieb lattice has a simple cubic symmetry, which ischaracterized by the single Dirac-cone per Brillouin zone and the middle flat band in theband structure. The intrinsic SOC is essentially needed to open the full energy gap in thebulk. The QSH effect could survive even in the presence of the exchange field. In terms ofthe first Chern number and the spin Chern number, we study the topological nature and thetopological phase transition from the time-reversal symmetry broken QSH effect to the QAHeffect. For Lieb lattice ribbons, the energy spectrum and the wave-function distributionsare obtained numerically, where the helical edge states and the chiral edge states revealthe non-trivial topological QSH and QAH properties, respectively.     

Modulation of Spin Distribution and Spin Transport by a Magnetic Field in a Quasi-One-Dimensional System with Spin--Orbit Coupling

The distributions of spin and currents modulated by magnetic field in a transverse parabolic confined two-dimensional electronic system with a Rashba spin--orbit coupling have been studied numerically. It is shown that the spin accumulation and the spin related current are generated by magnetic field if the spin--orbit coupling is presented. The distributions of charge and spin currents are antisymmetrical along the cross-section of confined system. A transversely applied electric field does not influence the characteristic behaviour of charge- and spin-dependent properties.     

Quantum spin Hall and quantum valley Hall effects in trilayer graphene and their topological structures

The present study pertains to the trilayer graphene in the presence of spin orbit coupling to probe the quantum spin/valley Hall effect. The spin Chern-number C_s for energy-bands of trilayer graphene having the essence of intrinsic spin–orbit coupling is analytically calculated. We find that for each valley and spin, C_s is three times larger in trilayer graphene as compared to single layer graphene. Since the spin Chern-number corresponds to the number of edge states,consequently the trilayer graphene has edge states, three times more in comparison to single layer graphene. We also study the trilayer graphene in the presence of both electric-field and intrinsic spin–orbit coupling and investigate that the trilayer graphene goes through a phase transition from a quantum spin Hall state to a quantum valley Hall state when the strength of the electric field exceeds the intrinsic spin coupling strength. The robustness of the associated topological bulk-state of the trilayer graphene is evaluated by adding various perturbations such as Rashba spin–orbit(RSO) interaction αR, and exchange-magnetization M. In addition, we consider a theoretical model, where only one of the outer layers in trilayer graphene has the essence of intrinsic spin–orbit coupling, while the other two layers have zero intrinsic spin–orbit coupling.Although the first Chern number is non-zero for individual valleys of trilayer graphene in this model, however, we find that the system cannot be regarded as a topological insulator because the system as a whole is not gaped.     

Novel topological insulators from crystalline symmetries

We discuss recent advances in the study of topological insulators protected by spatial symmetries by reviewing three representative, theoretical examples. In three dimensions (3D), these states of matter are generally characterized by the presence of gapless boundary states at surfaces that respect the protecting spatial symmetry. We discuss the appearance of these topological states in both crystals with negligible spin–orbit coupling and a fourfold rotational symmetry, as well as in mirror-symmetric crystals with sizable spin–orbit interaction characterized by the so-called mirror Chern number. Finally, we also discuss similar topological crystalline states in one-dimensional (1D) insulators, such as nanowires or atomic chains, with mirror symmetry. There, the prime physical consequence of the non-trivial topology is the presence of quantized end charges.     

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.     

Higher-Order Topological Spin Hall Effect of Sound

We theoretically propose a reconfigurable two-dimensional(2 D) hexagonal sonic crystal with higher-order topology protected by the six-fold,C₆,rotation symmetry.The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit cell.In the nontrivial phase,the sonic crystal realizes the topological spin Hall effect in a higher-order fashion:(i) the edge states emerging in the bulk band gap exhibit partial spin-momentum correlation and are gapp...     

Band topology and the quantum spin Hall effect in bilayer graphene

We consider bilayer graphene in the presence of spin-orbit coupling, in order to assess its behavior as a topological insulator. The first Chern number n for the energy bands of single-layer graphene and that for the energy bands of bilayer graphene are computed and compared. It is shown that for a given valley and spin, n for a Bernal-stacked bilayer is doubled with respect to that for the monolayer. This implies that this form of bilayer graphene will have twice as many edge states as single-layer graphene, which we confirm with numerical calculations and analytically in the case of an armchair terminated surface. Bernal-stacked bilayer graphene is a weak topological insulator, whose surface spectrum is susceptible to gap opening under spin-mixing perturbations. We assess the stability of the associated topological bulk state of bilayer graphene under various perturbations. In contrast, we show that AA-stacked bilayer graphene is not a topological insulator unless the spin-orbit coupling is bigger than the interlayer hopping. Finally, we consider an intermediate situation in which only one of the two layers has spin-orbit coupling, and find that although individual valleys have non-trivial Chern numbers for the case of Bernal stacking, the spectrum as a whole is not gapped, so the system is not a topological insulator.     

Emergent Z_2 topological invariant and robust helical edge states in two-dimensional topological metals

In this work, we study the effects of disorder on topological metals that support a pair of helical edge modes deeply embedded inside the gapless bulk states. Strikingly, we predict that a quantum spin Hall(QSH) phase can be obtained from such topological metals without opening a global band gap. To be specific, disorder can lead to a pair of robust helical edge states which is protected by an emergent Z_2 topological invariant, giving rise to a quantized conductance plateau in transport measurements. These results are instructive for solving puzzles in various transport experiments on QSH materials that are intrinsically metallic. This work also will inspire experimental realization of the QSH effect in disordered topological metals.     

Topological states in normal and superconducting p-wave chains

We study a two-band model of fermions in a 1d chain with an antisymmetric hybridization that breaks inversion symmetry. We find that for certain values of its parameters, the sp-chain maps formally into a p-wave superconducting chain, the archetypical 1d system exhibiting Majorana fermions. The eigenspectra, including the existence of zero energy modes in the topological phase, agree for both models. The end states too share several similarities, such as the behavior of the localization length, the non-trivial topological index and robustness to disorder. However, we show that the excitations in the ends of a finite sp chain are conventional fermions though endowed with protected topological properties. Our results are obtained by a scattering approach in a semi-infinite chain with an edge defect treated within the T-matrix approximation. We present exact numerical diagonalization results that extend our analysis to arbitrary parameters and to disordered systems.     

Spin current in a two-terminal quantum ring

The photo-induced direct current in a system consisting of a one-dimensional Rashba ring with conductors connected to it is calculated. The current is produced under the action of circularly polarized radiation incident on the ring in the presence of Rashba spin—orbit coupling. The charge and spin currents in conductors are expressed in terms of the coefficients of the electron transmission through the ring with inelastic interaction with radiation taken into account. It is shown that the spin current is a complex function of the magnetic flux through the ring, of the radiation frequency, and the spin—orbit coupling constant.     

Spin splitter regime of a mesoscopic Rashba ring

Using the non-equilibrium Greens' function formalism we calculate the spin currents in a one-dimensional ring coupled to three leads and in the presence of perpendicular magnetic flux Φ and Rashba spin-orbit coupling. A finite bias is applied between the input lead and the other two output leads. We show that the spin-orbit coupling allows one to operate this system as a spin splitter, i.e. the output leads deliver spin-polarized currents with different orientations. We find that the spin splitter operation can be tuned at integer multiples of Φ/Φ₀. Its efficiency depends not only on the value of the Rashba coupling but also on the bias applied between the input and output leads. The selected spin orientation of the output leads can be reversed by a slight change of their contact position. We discuss as well the connection between the spin splitter operation and the spectral properties of the ring.