Protected Edge Research Articles

Stably protected gapless edge states without Wannier obstruction

Bulk-boundary correspondence serves as an important feature of topological insulators, including Chern insulators and ${\mathbb{Z}}_{2}$ topological insulators. These topological insulators have spectral flow in the Wilson-loop spectrum, and they encounter Wannier obstruction. Recent studies show that the bulk topological features may not imply the existence of protected gapless boundary states. Here we address the opposite question: Does the existence of stably protected gapless edge states necessarily imply Wannier obstruction or Wilson-loop winding? We provide an example in which stably protected gapless edge states arise without the aforementioned bulk topological features. This trivialized topological insulator belongs to a new class of systems with non-$\ensuremath{\delta}$-like Wannier functions. Interestingly, the gapless edge states are not protected by crystalline symmetry; instead, the protection originates from mirror antisymmetry, a combination of chiral and mirror symmetries. Although the protected gapless edge states cannot be captured by bulk topological features, they can be characterized by the spectral flow in the entanglement spectrum.

Unidirectional spin-wave edge modes in magnonic crystal

We present a numerical demonstration of magnonic crystals hosting unidirectional, topologically protected edge states. The magnonic crystal is formed of dipolarly coupled Permalloy triangles. We show that due to the geometry of the block, the size of the structure can be scaled up. In addition, edge states can be found over a wide frequency range. Experimental detection of edge excitations in the considered system can be done with state-of-the-art techniques. Thus, we demonstrate a proof-of-concept magnonic Chern topological insulator nanostructure with simple geometry feasible for experimental realization. Furthermore, by tuning the strength of the perpendicular magnetic field, we induce a topological phase transition, which results in the change of direction of the topological edge state. Then, we demonstrate the magnonic switch based on this effect.

High-frequency topological corner and edge states in elastic honeycomb plates

Higher-order topological insulators (HOTIs) have attracted considerable attention because of their topological properties of low-dimensional topological corner states and edge states that exist in bandgaps. High-frequency edge states and corner states can be realized by controlling the interface configuration. However, most studies on HOTI focus on the bandgap at low frequencies, and the valley-selectivity of corner states is generally realized through a big-angle rotation. Here, we propose an elastic valley phononic crystal plate with an out-of-plane wide band gap to realize the edge states and corner states in the high-frequency bandgap. The sliding displacement of the resonators is tuned to control the topological phase transition. The rainbow trapping of topologically protected edge waves (TPEWs) is realized through the modulation of group velocities in the gradient structure based on the sliding parameter, and the valley-selectivity of corner states at high frequencies is further explored by engineering positions of the valley polarization. The simulation and experimental results indicate that TPEWs can be transmitted to different locations according to the switch of frequencies and corner states still hold valley-selectivity in high-frequency bandgaps. The proposed honeycomb plates provide generality for realizing high-frequency edge states and corner states and broaden the application for vibration transmission control and elastic energy trapping.

Study of bulk carrier properties in largely hybridized InAsSb/GaSb superlattice materials

The prospect of implementing quantum device architectures with technologically mature III–V semiconductors requires precisely controlled topologically protected edge states and bulk insulation. However, experimentally reaching this regime with III–V heterostructure epitaxy has been difficult due to charged bulk defects. Charged defects prevent bulk insulation and disturb the stability of edge states. Hence, we study carrier properties in a hybridized III–V InAs0.9Sb0.1/GaSb superlattice (SL) structure. We realize that an electron density (mobility) is limited to approximately 1012 cm−2 (104 cm2/V s). In order to understand these limits, the authors investigate the in-plane mobility of hybridized SLs as a function of current-carrying layer thickness, L, to determine scattering mechanisms that restrict carrier mobility. Although theory predicts the in-plane mobility is proportional to L6 dominated by interface roughness scattering (IRS) at low temperatures, we report that mobility follows the fourth power dependence, which is a weaker than expected from IRS theory. We attribute the discrepancy between experiment and the model to interface intermixing and wave function penetration into barrier regions. We use this understanding to develop a strategy for realizing high-performance topological materials.

WAR CRIMES IN GAZA STRIP FROM YEAR 2008 2021: INDIVIDUAL CRIMINAL RESPONSIBILITY UNDER THE LEGAL FRAMEWORK OF ROME STATUTE OF THE INTERNATIONAL CRIMINAL COURT

From the end of the year 2008 to 2021, Israel committed four wars in the Gaza Strip. These wars were known as Operation Cast Lead in 2009, Operation Pillar of Cloud in 2012, Operation Protective Edge in 2014, and Operation Guardian of the Walls in 2021. The destructive impacts on vital facilities in Gaza, including schools, universities, mosques, hospitals, and United Nations Relief centers, as well as the killings of thousands of innocent civilians by Israeli forces, are visible evidence of a serious violation of the principle of distinction, a basic principle provided under the international humanitarian law conferring protection upon the civilians during armed conflicts. The accession of Palestine to the Rome Statute of the International Criminal Court on crimes. This paper aims to analyze the position of individual criminal responsibility committed during the above-mentioned wars, regulated under the Rome Statute of the International Criminal Court and other relevant international conventions. In analyzing the data from primary and secondary sources, this paper applied doctrinal legal research and qualitative approaches. This paper concludes that the Rome Statute of the International Criminal Court associated the individual criminal responsibility for war crimes by systematically determining the extent of participation, namely: (1) commission; (2) ordering/instigating; (3) assistance; and (4) contribution to a war crime as a group crime. Thus, Israeli military leaders ought to be held liable for the commission of war crimes in accordance with Article 8 of the Rome Statute of the International Criminal Court. The pretexts of the Israeli occupation to be exempt from criminal responsibility should be rejected according to the state of legitimate defense.

OCCUPATION VS. RESISTANCE

The indiscriminate killing of Palestinians in the occupied Gaza Strip by the Israeli war machine, during July–August 2014, marked yet another phase in the long-drawn-out Palestinian resistance against Israeli occupation. The large-scale death, destruction, and displacement on the Palestinian side underlined a particular pattern of Israeli State behavior vis-à-vis the Palestinians in Gaza. This article argues that the 2014 operation, code named “Operation Protective Edge,” was but a part of Israel’s long-term, well-thought-out, and consistently pursued policy of crushing Palestinian resistance and eliminating Palestinian identity and nationalism.

Coexistence of quantum spin and valley hall effect in two-dimensional dielectric photonic crystals

The location and transmission of light is the core of modern photonic integrated device, and the proposal of topological photonics provides a new way of implementing optical manipulation. Topological photonic structures based on the quantum spin hall effect or quantum valley hall effect have the properties of immunity to defects and suppress backscattering, so they play a key role in designing novel low-loss photonic devices. In this work, we design a two-dimensional dielectric photonic crystal with time-reversal symmetry to achieve the coexistence of the quantum spin hall effect and the quantum valley hall effect in a photonic crystal. The design can be likened to an electronic system in which two pairs of Kramers simplex pairs are constructed to achieve a quadruple simplex pair in a photonic crystal. First, based on the method of shrinking and expanding the silicon pillars arranged in the honeycomb structure, the quadruple degeneracy point at the <i>Γ</i> point of the first Brillouin zone is opened, and the corresponding topologically trivial or non-trivial photonic band gap is formed,thereby realizing quantum spin hall effect. The expanded honeycomb lattice evolves into a Kagome structure, and then positive and negative perturbations are added to the Kagome lattice, breaking the spatial inversion symmetry of the Photonic crystal. When mirror symmetry is broken, different chiral photonic crystals can be created,leading the degeneracy point of the non-equivalent valleys <i>K</i> and <i>K'</i> in the Brillouin zone to be opened and a complete band gap to appear, thus realizing the Quantum valley hall effect. In the common band gap, topologically protected edge states are induced by nontrivial valley Chern number at the interface between two photonic crystals with opposite chirality. The numerical calculations show that unidirectional transport and bending-immune topological boundary states can be realized at the interface composed of topologically trivial (non-trivial) and positively (negatively) perturbed photonic crystals. Finally, a four-channel system based on the coexistence of the two effects is designed, The system is a novel electromagnetic wave router that can be selectively controlled by pseudospin degree of freedom or valley degree of freedom. This system provides a potential method for realizing the optical encoding and robust signal transmission, thereby providing greater flexibility for manipulating electromagnetic waves.

Switchable large-gap quantum spin Hall state in the two-dimensional MSi2Z4 class of materials

Quantum spin Hall (QSH) insulators exhibit spin-polarized conducting edge states that are topologically protected from backscattering and offer unique opportunities for addressing fundamental science questions and device applications. Finding viable materials that host such topological states, however, remains a challenge. Here by using in-depth first-principles theoretical modeling, we predict large bandgap QSH insulators in recently bottom-up synthesized two-dimensional (2D) MSi$_2$Z$_4$ (M = Mo or W and Z = P or As) materials family with $1T^\prime$ structure. A structural distortion in the $2H$ phase drives a band inversion between the metal (Mo/W) $d$ and $p$ states of P/As to realize spinless Dirac cone states without spin-orbit coupling. When spin-orbit coupling is included, a hybridization gap as large as $\sim 204$ meV opens up at the band crossing points, realizing spin-polarized conducting edge states with nearly quantized spin Hall conductivity. We also show that the inverted band gap is tunable with a vertical electric field which drives a topological phase transition from the QSH to a trivial insulator with Rashba-like edge states. Our study identifies 2D MSi$_2$Z$_4$ materials family with $1T^\prime$ structure as large bandgap, tunable QSH insulators with protected spin-polarized edge states and large spin-Hall conductivity.

An epitaxial graphene platform for zero-energy edge state nanoelectronics

Graphene’s original promise to succeed silicon faltered due to pervasive edge disorder in lithographically patterned deposited graphene and the lack of a new electronics paradigm. Here we demonstrate that the annealed edges in conventionally patterned graphene epitaxially grown on a silicon carbide substrate (epigraphene) are stabilized by the substrate and support a protected edge state. The edge state has a mean free path that is greater than 50 microns, 5000 times greater than the bulk states and involves a theoretically unexpected Majorana-like zero-energy non-degenerate quasiparticle that does not produce a Hall voltage. In seamless integrated structures, the edge state forms a zero-energy one-dimensional ballistic network with essentially dissipationless nodes at ribbon–ribbon junctions. Seamless device structures offer a variety of switching possibilities including quantum coherent devices at low temperatures. This makes epigraphene a technologically viable graphene nanoelectronics platform that has the potential to succeed silicon nanoelectronics.

Complex dispersion relation and topologically protected flexural wave transport in viscoelastic periodic plates

Periodic plates are regarded as a promising platform for topological insulator investigation. However, most studies on the topological insulator are conducted by the ω(k) method and the material viscosity is usually ignored. The weakness of the ω(k) method is that the imaginary part of wave vector (evanescent wave modes) cannot be obtained. In other words, the attenuation property cannot be revealed. On the other hand, the material viscosity has a major effect on the dynamic responses of periodic structures, and cannot be considered in the ω(k) method also. To overcome these limitations, the complex dispersion method or the k(ω) method based on FEM is developed here and extended to the investigation of topologically protected flexural wave transport in viscoelastic periodic plates. In detail, we propose a double-sided stubbed periodic plate, mimicking the quantum valley Hall effect. The complex dispersion relation is obtained that reveals all the wave modes of the present periodic plates, exhibiting the propagative and attenuation properties comprehensively. Through the combination of complex dispersion analysis and full-field simulation, the evolution of dispersion relation regarding to material damping is discovered. Topological wave transport is worsened up to stop by the material viscosity. However, the robustness and energy localized capacity of the topologically protected edge states is little affected by material viscosity under the condition of small loss. This study provides a new perspective for analyzing topologically protected edge states and promotes the practical application of topological insulator.

Electronic and topological properties of extended two-dimensional Su-Schrieffer-Heeger models and realization of flat edge bands

One-dimensional Su-Schrieffer-Heeger (SSH) chains are one of the simplest topological models and have been studied extensively. The two-dimensional $(2\mathrm{D})$ version has been confirmed to have a nontrivial topology. In this work, we further extend the $2\mathrm{D}$ SSH model by constructing different configurations, including all possible configurations with 4-site unit cells and another two complicated, but typical, configurations with 8- and 36-site unit cells, respectively. We calculate and analyze the electronic structures and topologies of these SSH models in detail and identify several rich and novel properties, such as topologically protected edge states, metallic chains with different shapes, and a bulk--edge separation in a metal system. In particular, a flat-band feature of topological edge states is obtained. By analyzing the spatial distribution of these edge states, we explain the origin of the flat edge bands and show that bonding squares play a crucial role in the formation of flat bands. Our study can be generalized to more configurations and higher dimensions, providing a basis for further theoretical and experimental explorations.

Combat Injury Profile in Urban Warfare.

Combat ground maneuvers consist of various platforms and have several environmental characteristics, influenced by the terrain, the operational mission, and the force's capabilities. This study assesses data on injuries sustained during urban warfare, aiming to evaluate the relationship between injury characteristics, maneuver platform, and personal protective gear on the battlefield. IDF soldiers injured infantry soldiers from the "Cast Lead" and the "Protective Edge" operations in the Gaza Strip (2008-2009 and 2014, respectively) were divided into four groups according to the maneuver platform and the environment: mounted infantry (armored and unarmored vehicle) and dismounted infantry (urban and open area). The primary outcome was the severity of the injury, and the secondary outcome was the injured body part. Overall, 588 casualties were included in the final analysis, of whom 507 were dismounted infantry soldiers (265 in open terrain and 242 in urban area) and 81 were mounted infantry soldiers (20 in unarmored and 61 were injured in armored vehicles). The Injury Severity Score was similar in all subgroups. Open terrain subgroups were found to have fewer head injuries and higher levels of lower extremity injuries, similar to the unarmored vehicle group. More facial injuries were documented in the urban area group. The Injury Severity Score was not influenced by environmental protection. Although we found differences in the injured body parts, further studies on the exact mechanism of injury are needed to elucidate further the relationship and differences between the various platforms used and injuries seen in urban warfare, aiming for tailor-made protection.

All-Optically Controlled Topological Current Switcher Based on Silicene-Like Nanoribbons

Low-dimensional topological insulator materials, such as silicene-like 2-D materials, have wide applications in electronic, spintronic, and optoelectronic nanodevice, by utilizing their topologically protected edge states. In theoretical studies, Floquet theorem is usually employed in the nanosystems under light irradiation with high-frequency approximation, which may not be accessible in experimental research. In this work, we study the quasienergy spectra and edge-state transport of silicene-like nanoribbons under low-frequency light irradiation (in visible wavelengths), by extracting the zero-photon component from the nonperturbative Floquet bands. These silicene-like nanoribbons undergo topological phase transition from quantum spin Hall (QSH) to quantum anomalous Hall (QAH) insulator under circularly polarized light, accompanied by helical edge modes to chiral ones. The propagation direction of edge-mode current in QAH nanoribbons can be switched by left-/right-circularly polarized (LCP/RCP) light, and such a current-switching effect is then confirmed in heterojunction composed of QSH and QAH nanoribbons. Based on the all-optically controlled current-switching effect, we design a three-terminal current switcher, with one input terminal, two output ones, and central device irradiated by LCP/RCP light. Conductance and real-space local current distribution of the proposed nanosystem reveal that the input current can indeed be switched into specific output terminal by LCP/RCP light. The proposed current switcher changes the electron movement very quickly, and we believe that it would have potential applications in high-speed photoelectric nanodevices.

Large Gap Topological Insulating Phase and Anisotropic Rashba and Chiral Spin Textures in Monolayer Zintl A2MX2

A family of two-dimensional (2D) materials, particularly the complex-structured Zintl phase compounds, has attracted tremendous research attention because of tunable material properties and exceptional applications. Utilizing first-principles computations under the hybrid functional approach, we performed a systematic study on A2MX2 (A = Ca, Sr, or Yb; M = Zn or Cd; X = P, As, Sb, or Bi). Among the three surface terminations considered, the metal element XM-termination (T2) was found to be the most stable structural phase with the lowest total ground state energies. Thermodynamic stability was further confirmed through phonon dispersion and formation energy calculations. Surprisingly, the T2 monolayer Sr2CdBi2 was found to host the topological insulating phase with a sizable bandgap of 556 meV, as well as a Z2 topological invariance of 1. The corresponding topologically protected edge states link the conduction and valence bands. Moreover, the topological phase is driven by the spin–orbit coupling, which causes the inverted bands at Γ point close to the Fermi level concerning the Cd-s + Bi2-s and Bi2-px + py orbitals. Moreover, Rashba and chiral spin-splittings were also observed. The computed Rashba strengths along Γ-M (αRΓ-M) are 1.970, 0.676, and 0.669 eV·Å, whereas the computed values along Γ-K (αRΓ-K) are 1.701, 0.731, and 0.646 eV·Å, for Ca2ZnAs2, Sr2CdBi2, and Yb2ZnAs2, respectively. Our study provides fundamental knowledge to further experimental investigations and synthesis, which may lead to electronic applications of A2MX2 compounds in quantum computing or spintronics.

Factors influencing lion movements and habitat use in the western Serengeti ecosystem, Tanzania

Protected areas that restrict human activities can enhance wildlife habitat quality. Efficacy of protected areas can be improved with increased protection from illegal activities and presence of buffer protected areas that surround a core protected area. Habitat value of protected areas also can be affected by seasonal variation in anthropogenic pressures. We examined seasonal space use by African lions (Panthera leo) within a core protected area, Serengeti National Park, Tanzania, and surrounding buffer protected areas with varying protection strengths. We used lion locations in logistic regression models during wet and dry seasons to estimate probability of use in relation to protection strength, distance to protected area edge, human and livestock density, distance to roads and rivers, and land cover. Lions used strongly protected buffer areas over the core protected area and unprotected areas, and moved away from protected area boundaries toward the core protected area when buffer protected areas had less protection. Lions avoided high livestock density in the wet season and high human density in the dry season. Increased strength of protection can decrease edge effects on buffer areas and help maintain habitat quality of core protected areas for lions and other wildlife species.

Topological Polarization Beam Splitter in Dual-Polarization All-Dielectric Valley Photonic Crystals

The recent realization of all-dielectric valley photonic crystals (VPCs) in nanoscale not only provides a standard platform to explore many topological phases of light with protected edge modes, but also shows promising applications for designing high-performance nanophotonic devices. However, the widely reported all-dielectric VPCs are limited to one single polarization, and it is challenging to manipulate the polarization degree of freedom of light and design polarizing devices using VPCs. Here, we design a dual-polarization all-dielectric VPC and propose a topological polarization beam splitter. The phase vortex distributions of bulk modes at different valleys characterize the nonzero valley Chern numbers for both TE and TM polarizations, indicating the implementation of topologically nontrivial dual-polarization band gaps. This leads to dual-polarization valley-dependent edge modes located at interfaces with different shapes. The topological valley transport around sharp-bends is also demonstrated for both TE and TM polarizations. Finally, a harpoon-shaped polarization beam splitter is proposed, and the simulated results confirm the good functionality of polarization separation. Our work shows the flexible control of light in topological photonic systems with the polarization degree of freedom, and has promising applications in polarization multiplexing photonic devices owing to the enlarged the optical information-processing capacity.

Simultaneous slow light and sound rainbow trapping in phoxonic crystals.

In this paper, we use a phoxonic crystal (PxC) which can control the topological states of light and sound by breaking inversion symmetry and thus make it possible to achieve rainbow trapping of light and sound simultaneously. It is shown that topologically protected edge states can be obtained at the interfaces between PxCs with different topological phases. Thus, we designed a gradient structure to realize the topological rainbow trapping of light and sound by linearly modulating the structural parameter. In the proposed gradient structure, the edge states of light and sound modes with different frequencies are respectively trapped at different positions, owing to near zero group velocity. The topological rainbows of light and sound are simultaneously realized in one structure, which open a new, to the best of our knowledge, view and provide a feasible platform for the application of the topological optomechanical devices.

Topological valley crystals in a photonic Su–Schrieffer–Heeger (SSH) variant

Progress on two-dimensional materials has shown that valleys, as energy extrema in a hexagonal first Brillouin zone, provide a new degree of freedom for information manipulation. Then, valley Hall topological insulators supporting such-polarized edge states on boundaries were set up accordingly. In this paper, a two-dimensional valley crystal composed of six tunable dielectric triangular pillars in each unit cell is proposed in the photonic sense of a deformed Su–Schrieffer–Heeger model. We reveal the vortex nature of valley states and establish the selection rules for valley-polarized states. Based on the valley topology, a rhombus-shaped beam splitter waveguide is designed to verify the valley-chirality selection rule above. Our numerical results entail that this topologically protected edge states still maintain robust transmission at sharp corners, thus providing a feasible idea for valley photonic devices in the THz regime.

Electrical control of spin-polarized topological currents in monolayer WTe2

We evidence the possibility for coherent electrical manipulation of the spin orientation of topologically protected edge states in a low-symmetry quantum spin Hall insulator. By using a combination of ab initio simulations, symmetry-based modeling, and large-scale calculations of the spin Hall conductivity, it is shown that small electric fields can efficiently vary the spin textures of edge currents in monolayer 1T'-${\mathrm{WTe}}_{2}$ by up to a 90-degree spin rotation, without jeopardizing their topological character. These findings suggest a new kind of gate-controllable spin-based device, topologically protected against disorder and of relevance for the development of topological spintronics.

Analytic expressions for topologically protected edge states in Su–Schrieffer–Heeger model

The Su–Schrieffer–Heeger (SSH) model can demonstrate the topological phase transition of spinless fermions in a two-dimensional square lattice. In this work, we investigate the topologically protected edge states (TESs) in 1D SSH ribbons by using the wave mechanics approach. A concise analytic equation is derived to determine the localization length of TESs. Based on the resulted localization length, the explicit analytic expressions are given for the energy dispersions and wave functions of TESs in 1D SSH ribbons. The effect of the size of the 1D SSH ribbon on the topological phase transition and the critical exponent of the localization length of TESs are further demonstrated. It also shows that the degeneracy of TESs is lifted due to the coupling of the two-side TESs. Our results can elucidate the TESs appearing in nontrivial phase, and can also be employed to optimize the structure of the nanoribbon devices.