Types Of Edge States Research Articles

Topological edge and corner states in coupled wave lattices in nonlinear polariton condensates

Abstract Topological states have been widely investigated in different types of systems and lattices. In the present work, we report on topological edge states in double-wave (DW) chains, which can be described by a generalized Aubry-André-Harper (AAH) model. For the specific system of a driven-dissipative exciton polariton system we show that in such potential chains, different types of edge states can form. For resonant optical excitation, we further find that the optical nonlinearity leads to a multistability of different edge states. This includes topologically protected edge states evolved directly from individual linear eigenstates as well as additional edge states that originate from nonlinearity-induced localization of bulk states. Extending the system into two dimensions (2D) by stacking horizontal DW chains in the vertical direction, we also create 2D multi-wave lattices. In such 2D lattices multiple Su–Schrieffer–Heeger (SSH) chains appear along the vertical direction. The combination of DW chains in the horizonal and SSH chains in the vertical direction then results in the formation of higher-order topological insulator corner states. Multistable corner states emerge in the nonlinear regime.

Effects of topological and non-topological edge states on information propagation and scrambling in a Floquet spin chain

The action of any local operator on a quantum system propagates through the system carrying the information of the operator. This is usually studied via the out-of-time-order correlator (OTOC). We numerically study the information propagation from one end of a periodically driven spin-1/2 XY chain with open boundary conditions using the Floquet infinite-temperature OTOC. We calculate the OTOC for two different spin operators, σ x and σ z . For sinusoidal driving, the model can be shown to host different types of edge states, namely, topological (Majorana) edge states and non-topological edge states. We observe a localization of information at the edge for both σ z and σ x OTOCs whenever edge states are present. In addition, in the case of non-topological edge states, we see oscillations of the OTOC in time near the edge, the oscillation period being inversely proportional to the gap between the Floquet eigenvalues of the edge states. We provide an analytical understanding of these effects due to the edge states. It was known earlier that the OTOC for the spin operator which is local in terms of Jordan–Wigner fermions (σ z ) shows no signature of information scrambling inside the light cone of propagation, while the OTOC for the spin operator which is non-local in terms of Jordan–Wigner fermions (σ x ) shows signatures of scrambling. We report a remarkable ‘unscrambling effect’ in the σ x OTOC after reflections from the ends of the system. Finally, we demonstrate that the information propagates into the system mainly via the bulk states with the maximum value of the group velocity, and we show how this velocity is controlled by the driving frequency and amplitude.

Topological Edge States on Different Domain Walls of Two Opposed Helical Waveguide Arrays

Floquet topological insulators (FTIs) have richer topological properties than static systems. In this work, we designed different domain wall (DW) structures consisting of a Floquet photonic lattice with opposite helical directions. We investigated the existence and types of edge states in three shared coupling structures and the impact of these shared coupling structures on edge states. When two opposite helical lattices share a straight waveguide array coupling, the edge states are localized on the straight waveguide. When two opposite helical lattices share a clockwise (or anticlockwise) helical waveguide array coupling, the DWs consist of zigzag and bearded edges, but the positions of the zigzag and bearded edges of the shared clockwise waveguide array are different from those of the shared anticlockwise waveguide array. The slope and transmission rate of the edge states both vary with the degree of coupling between the shared waveguides. The characteristics of these edge states, such as transmission speed and band gap width, are also affected by the incidence angle, modulation phase factor, and helical radii, and the methods for controlling the edge states in different shared coupling structures are provided. This will help deepen our understanding of how topological structures influence the electronic and photonic properties of materials. This could also lead to combining topology with metasurface-based structured light, which would highlight many novel properties with great application potential for various fields, such as imaging, metrology, communication, quantum information processing, and light–matter interaction.

Realization of Photonic Topological Insulators at Terahertz Frequencies Characterized by Time-Domain Spectroscopy

Topological systems are inherently robust to disorder and continuous perturbations, resulting in dissipation-free edge transport of electrons in quantum solids, or reflectionless guiding of photons and phonons in classical wave systems characterized by topological invariants. Realization of robust, lossless topological insulators in the technologically important terahertz (THz) frequency range has drawn immense attention. In this paper, we propose a Kagome lattice structure that supports topological edge states with valley-dependent transport. Optical topological insulators are subsequently fabricated, in which mirror symmetry is broken and degenerate states are lifted at $K({K}^{\ensuremath{'}})$ valleys in the band structure. Theoretical and numerical analyses show that four types of edge states can be obtained. The performance of the fabricated topological waveguides is characterized with THz time-domain spectroscopy (TDS). Nearly identical THz wave transmission in the 0.437--0.453-THz range is verified with a straight-line topological waveguide and Z-shaped waveguides with multiple sharp turning corners, which quantitatively illustrate the strong backscattering suppression effects from the nontrivial topology edge states of the structures. Particularly, the THz TDS measurements provide important extra time-domain information about the topological waveguiding mode, which is generally unavailable from previous topological insulator investigations solely replying on power spectrum measurements.

Phase diagrams and edge-state transitions in graphene with spin-orbit coupling and magnetic and pseudomagnetic fields

The quantum Hall (QH) effect, the quantum spin Hall (QSH) effect, and the quantum valley Hall (QVH) effect are three peculiar topological insulating phases in graphene. They are characterized by three different types of edge states. These three effects are caused by the external magnetic field, the intrinsic spin-orbit coupling (SOC) and the strain-induced pseudomagnetic field, respectively. Here we theoretically study phase diagrams when these effects coexist and analyze how the edge states evolve between the three. We find the real magnetic field and the pseudomagnetic field will compete above the SOC energy gap while the QSH effect is almost unaffected within the SOC energy gap. The edge-state transitions from the QH effect or the QVH effect to the QSH effect directly relies on the arrangement of the zeroth Landau levels. Using edge-state transitions, we raise a device similar to a spin field effect transistor (spin-FET) and also design a spintronics multiple-way switch.

Edge states in proximitized graphene ribbons and flakes in a perpendicular magnetic field: Emergence of lone pseudohelical pairs and pure spin-current states

We investigate the formation of edge states in graphene ribbons and flakes with proximity-induced valley-Zeeman and Rashba spin-orbit couplings in the presence of a perpendicular magnetic field $B$. Two types of edge states appear in the spin-orbit gap at the Fermi level at zero field: strongly localized pseudohelical (intervalley) states and weakly localized intravalley states. We show that if the magnetic field is stronger than a crossover field ${B}_{c}$, which is a few milliteslas for realistic systems such as $\mathrm{graphene}/\mathrm{W}{\mathrm{Se}}_{2}$, only the pseudohelical edge states remain in zigzag graphene ribbons; the intravalley states disappear. The crossover is directly related to the closing and reopening of the bulk gap formed between nonzero Landau levels. Remarkably, in finite flakes the pseudohelical states undergo perfect reflection at the armchair edges if $B>{B}_{c}$, forming standing waves at the zigzag edges. These standing waves comprise two counterpropagating pseudohelical states; so while they carry no charge current, they do carry (pure) spin current.

Generalized Su-Schrieffer-Heeger Model in One Dimensional Optomechanical Arrays

We propose an implementation of a generalized Su-Schrieffer-Heeger (SSH) model based on optomechanical arrays. The topological properties of the generalized SSH model depend on the effective optomechanical interactions which can be controlled by strong driving fields. Three phases including one trivial and two distinct topological phases are found in the generalized SSH model. The phase transition can be observed by turning the strengths and phases of the effective optomechanical interactions via adjusting the driving fields. Moreover, four types of edge states can be created in generalized SSH model of an open chain under single-particle excitation, and the dynamical behaviors of the excitation in the open chain are related to the topological properties under the periodic boundary condition. We show that the edge states can be pumped adiabatically along the optomechanical arrays by periodically modulating the amplitude and frequency of the driving fields, and the state pumping is robust against small disorders. The generalized SSH model based on the optomechanical arrays provides us a controllable platform to engineer topological phases for photons and phonons, which may have potential applications in controlling the transport of photons and phonons.

Dual-band all-dielectric chiral photonic crystal

We present an all-dielectric chiral photonic crystal that guides the propagation of electromagnetic waves without backscattering for dual bands. The chiral photonic crystal unit cell is composed of four dielectric cylinders with increasing inner diameter clockwise or anticlockwise, which leads to chirality. It is demonstrated that the proposed chiral photonic crystal can generate dual band gaps in the gigahertz frequency range and has two types of edge states, which is similar to topologically protected edge states. Hence, the interface formed by the proposed 2D chiral photonic crystal can guide the propagation of electromagnetic waves without backscattering, and this complete propagation is immune to defects (position disorder or frequency disorder). To illustrate the applicability of the findings in communication systems, we report a duplexer and a power divider based on the presented all-dielectric chiral photonic crystal.

Living on the edge: Topology, electrostatics, and disorder

We address the co-existence of massless and massive topological edge states at the interface between two materials with different topological phases. We modify the well known Bernevig-Hughes-Zhang model to introduce a smooth function describing the band inversion and the band bending due to electrostatic effects between the bulk of the quantum well and the vacuum. Within this minimal model we identify distinct parameter sets that can lead to the co-existence of the two types of edge states, and that determine their number and characteristics. We propose several experimental setups that could demonstrate their presence in two-dimensional topological systems, as well as ways to regulate or tune the contribution of the massive edge states to the conductance of associated electronic devices. Our results suggest that such states may also be present in novel two-dimensional Van der Waals topological materials.

Tuning the bandgap and introducing magnetism into monolayer BC3 by strain/defect engineering and adatom/molecule adsorption

Using first-principles calculations, we study the structural, electronic, and optical properties of pristine BC3. Our results show that BC3 is a semiconductor which can be useful in optoelectronic device applications. Furthermore, we found that the electronic properties of BC3 can be modified by strain and the type of edge states. With increasing thickness, the indirect bandgap decreases from 0.7 eV (monolayer) to 0.27 eV (bulk). Upon uniaxial tensile strain along the armchair and zigzag directions, the bandgap slightly decreases, and with increasing uniaxial strain, the bandgap decreases, and when reaching −8%, a semiconductor-to-metal transition occurs. By contrast, under biaxial strain, the bandgap increases to 1.2 eV in +8% and decreases to zero in −8%. BC3 nanoribbons with different widths exhibit magnetism at the zigzag edges, while, at the armchair edges, they become semiconductor, and the bandgap is in the range of 1.0–1.2 eV. Moreover, we systematically investigated the effects of adatoms/molecule adsorption and defects on the structural, electronic, and magnetic properties of BC3. The adsorption of various adatoms and molecules as well as topological defects (vacancies and Stone-Wales defects) can modify the electronic properties. Using these methods, one can tune BC3 into a metal, half-metal, ferromagnetic-metal, and dilute-magnetic semiconductor or preserve its semiconducting character.

Floquet states of valley-polarized metal with one-way spin or charge transport in zigzag nanoribbons

Two-dimensional Floquet systems consisting of irradiated valley-polarized metal are investigated. For the corresponding static systems, we consider two graphene models of valley-polarized metal with either a staggered sublattice or uniform intrinsic spin-orbital coupling, whose Dirac point energies are different from the intrinsic Fermi level. If the frequency of irradiation is appropriately designed, the largest dynamical gap (first-order dynamical gap) opens around the intrinsic Fermi level. In the presence of the irradiation, two types of edge states appear at the zigzag edge of the semi-infinite sheet with energy within the first-order dynamical gap: the Floquet edge states and the strongly localized edge states. In narrow zigzag nanoribbons, the Floquet edge states are gapped out by the finite-size effect and the strongly localized edge states remain gapless. As a result, the conducting channels of the nanoribbons consist of the strongly localized edge states. Under the first and second model, the strongly localized edge states carry one-way spin-polarized and one-way charge current around the intrinsic Fermi level, respectively. Thus, the narrow zigzag nanoribbons of the first and second model have asymmetric spin and charge transmission rates, respectively. Quantum-transport calculations predict sizable pumped currents of charge and spin, which could be controlled by the Fermi level.

Topological nodal points in two coupled Su-Schrieffer-Heeger chains

We study two coupled Su-Schrieffer-Heeger (SSH) chains system, which is shown to contain rich quantum phases associated with topological invariants protected by symmetries. In the weak coupling region, the system supports two non-trivial topological insulating phases, characterized by winding number N = +1 or -1, and two types of edge states. The boundary between the two topological phases arises from two band closing points, which exhibit topological characteristics in one-dimensional k space. By mapping Bloch states on a vector field in k space, the band degenerate points correspond to a pair of kinks of the field, with opposite topological charges. Two topological nodal points move and merge as the inter-chain coupling strength varies. This topological invariant is protected by the translational and inversion symmetries, rather than the antiunitary operation. Furthermore, we find that when a pair of nodal points is created, a second order quantum phase transition (QPT) occurs, associating with a gap closing and spontaneously symmetry breaking. This simple model demonstrates several central concepts in the field of quantum materials and provides a theoretical connection between them.

Magnetic properties of graphene quantum dots

Using the tight-binding approximation we calculated the magnetic susceptibility of graphene quantum dots (GQD) of different geometrical shapes and sizes, smaller than the magnetic length, when the magnetic properties are governed by the electron edge states. Two types of edge states can be discerned: the zero-energy states (ZES) located exactly at the zero-energy Dirac point, and the dispersed edge states (DES) with the energy close, but not exactly equal to zero. DES are responsible for the temperature independent diamagnetic response, while ZES provide the temperature dependent spin Curie paramagnetism. The hexagonal, circular and randomly shaped GQDs contain mainly DES and, as a result, they are diamagnetic. The edge states of the triangular GQDs are ZES and these dots reveal the interplay between the spin paramagnetism, dominating for small dots and at low temperatures, and bulk orbital diamagnetism, dominating for large dots and at high temperatures.

Zigzag graphene nanoribbon edge reconstruction with Stone-Wales defects

In this article, we study zigzag graphene nanoribbons with edges reconstructed with Stone-Wales defects, by means of an empirical (first-neighbor) tight-binding method, with parameters determined by ab-initio calculations of very narrow ribbons. We explore the characteristics of the electronic band structure with a focus on the nature of edge states. Edge reconstruction allows the appearance of a new type of edge states. They are dispersive, with non-zero amplitudes in both sub-lattices; furthermore, the amplitudes have two components that decrease with different decay lengths with the distance from the edge; at the Dirac points one of these lengths diverges, whereas the other remains finite, of the order of the lattice parameter. We trace this curious effect to the doubling of the unit cell along the edge, brought about by the edge reconstruction. In the presence of a magnetic field, the zero-energy Landau level is no longer degenerate with edge states as in the case of pristine zigzag ribbon.

Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and edge states

There are two types of edge states in graphene with/without magnetic field. One is a quantum Hall edge state, which is topologically protected against small perturbation. The other is a chiral zero mode that is localized near the boundary with/without magnetic field. The latter is also topological but is guaranteed to be at zero energy by the chiral symmetry, which is also responsible for massless Dirac-like dispersion. Conceptual roles of the edge states are stressed and reviewed from the point of view of the bulk-edge correspondence and topological order.

Edge states and the quantized Hall effect in graphene

We study edges states of graphene ribbons in the quantized Hall regime, and show that they can be described within a continuum model (the Dirac equation) when appropriate boundary conditions are adopted. The two simplest terminations, zigzag and armchair edges, are studied in detail. For zigzag edges, we find that the lowest Landau level states terminate in two types of edge states, dispersionless and current-carrying surface states. The latter involve components on different sublattices that may be separated by distances far greater than the magnetic length. For armchair edges, the boundary conditions are met by admixing states from different valleys, and we show that this leads to a single set of edges states for the lowest Landau level and two sets for all higher Landau levels. In both cases, the resulting Hall conductance step for the lowest Landau level is half that between higher Landau levels, as observed in experiment.