Trivial Edge Research Articles

On the structure of vertex stabilizers of arc-transitive locally quasiprimitive graphs

Let Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document} be a connected arc-transitive G-graph which is locally finite and locally quasiprimitive. Let {x,y}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x,y\\}$$\\end{document} be an edge of Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}. A relation between Gx[1]/Op(Gx[1])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_x^{[1]}/O_p(G_x^{[1]})$$\\end{document} and the existence of certain normal subgroups of GxΔ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_x^{\\Delta (x)}$$\\end{document} and Gx,yΔ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_{x,y}^{\\Delta (x)}$$\\end{document} is established. This is then used to determine the vertex stabilizers of a class of 2-arc-transitive graphs with trivial edge kernel.

Introduction of Local Resonators to a Nonlinear Metamaterial With Topological Features

Abstract Recent work in nonlinear topological metamaterials has revealed many useful properties such as amplitude dependent localized vibration modes and nonreciprocal wave propagation. However, thus far, there have not been any studies to include the use of local resonators in these systems. This work seeks to fill that gap through investigating a nonlinear quasi-periodic metamaterial with periodic local resonator attachments. We model a one-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasi-periodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The dispersion relation of the infinite chain of the proposed system is determined analytically through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasi-periodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators. These results highlight the interplay between local resonance and nonlinearity in a topological metamaterial demonstrating for the first time the presence of an amplitude invariant bandgap alongside amplitude dependent topological bandgaps.

Quantum Transport of Topological and Trivial Edge States in Sn‐Doped SbH Nanoribbons

Although the transport of topological edge states may be insensitive to impurities, impurities may induce the interedge scattering and backscattering in topological nanoribbons due to the finite size effect. It may be interesting to compare the transport of topological edge states with that of trivial edge states in nanoribbons. Here, the quantum transport of edge states in the Sn‐doped SbH nanoribbons using the density functional theory combined with nonequilibrium Green's function method is studied. Both topological and trivial edge states can be found in the zigzag SbH nanoribbons. The impurity generates a higher barrier near the doping position and leads to the scattering of the states. From calculations, the transmission of topological edge states can be smaller than that of trivial edge states due to the antiresonance behavior. Compared with topological edge states, the transport of trivial edge states is more sensitive to the impurity position. Besides, the edge impurity is more likely to induce scattering for both topological and trivial edge states than the central impurity in narrow SbH nanoribbons, since the transport channels are limited to the edges. This study may be helpful for understanding the transport of edge states in nanoribbons.

Optical N-plasmon: topological hydrodynamic excitations in graphene from repulsive Hall viscosity

Edge states occurring in Chern and quantum spin-Hall phases are signatures of the topological electronic band structure in two-dimensional (2D) materials. Recently, a new topological electromagnetic phase of graphene characterized by the optical N-invariant was proposed. Optical N-invariant arises from repulsive Hall viscosity in hydrodynamic many-body electron systems, distinct from the Chern and Z 2 invariants. In this paper, we introduce the topologically protected edge excitation—optical N-plasmon of interacting many-body electron systems in the topological optical N-phase. These optical N-plasmons are signatures of the topological plasmonic band structure in 2D materials. We demonstrate that optical N-plasmons exhibit unique dispersion relations, stability against various boundary conditions, and edge profiles when compared with the topologically trivial edge magneto plasmons. Based on the optical N-plasmon, we design an ultra sub-wavelength broadband topological hydrodynamic circulator, which is a chiral quantum radio-frequency circuit component crucial for information routing and interfacing quantum–classical computing systems. Furthermore, we reveal that optical N-plasmons can be effectively tuned by the neighboring dielectric environment without breaking the topological properties. Our work provides a smoking gun signature of topological electromagnetic phases occurring in 2D materials arising from repulsive Hall viscosity.

Identifying s-wave pairing symmetry in single-layer FeSe from topologically trivial edge states

Determining the pairing symmetry of single-layer FeSe on SrTiO3 is the key to understanding the enhanced pairing mechanism. It also guides the search for superconductors with high transition temperatures. Despite considerable efforts, it remains controversial whether the symmetry is the sign-preserving s- or the sign-changing s±-wave. Here, we investigate the pairing symmetry of single-layer FeSe from a topological point of view. Using low-temperature scanning tunneling microscopy/spectroscopy, we systematically characterize the superconducting states at edges and corners of single-layer FeSe. The tunneling spectra collected at edges and corners show a full energy gap and a substantial dip, respectively, suggesting the absence of topologically non-trivial edge and corner modes. According to our theoretical calculations, these spectroscopic features can be considered as strong evidence for the sign-preserving s-wave pairing in single-layer FeSe.

Rotating topological edge solitons.

We address the formation of topological edge solitons in rotating Su-Schrieffer-Heeger waveguide arrays. The linear spectrum of the non-rotating topological array is characterized by the presence of a topological gap with two edge states residing in it. Rotation of the array significantly modifies the spectrum and may move these edge states out of the topological gap. Defocusing nonlinearity counteracts this tendency and shifts such modes back into the topological gap, where they acquire the structure of tails typical of topological edge states. We present rich bifurcation structure for rotating topological solitons and show that they can be stable. Rotation of the topologically trivial array, without edge states in its spectrum, also leads to the appearance of localized edge states, but in a trivial semi-infinite gap. Families of rotating edge solitons bifurcating from the trivial linear edge states exist too, and sufficiently strong defocusing nonlinearity can also drive them into the topological gap, qualitatively modifying the structure of their tails.

Optically controllable coupling between edge and topological interface modes of graphene metasurfaces

Nonlinear topological photonics has been attracting increasing research interest, as it provides an exciting photonic platform that combines the advantages of active all-optical control offered by nonlinear optics with the unique features of topological photonic systems, such as topologically-protected defect-immune light propagation. In this paper, we demonstrate that topological interface modes and trivial edge modes of a specially designed graphene metasurface can be coupled in a tunable and optically controllable manner, thus providing an efficient approach to transfer optical power to topologically protected states. This is achieved in a pump-signal configuration, in which an optical pump propagating in a bulk mode of the metasurface is employed to tune the band structure of the photonic system and, consequently, the coupling coefficient and wave-vector mismatch between edge and topological interface modes. This tunable coupling mechanism is particularly efficient due to the large Kerr coefficient of graphene. Importantly, we demonstrate that the required pump power can be significantly reduced if the optical device is operated in the slow-light regime. We perform our analysis using both ab initio full-wave simulations and a coupled-mode theory that captures the main physics of this active coupler and observe a good agreement between the two approaches. This work may lead to the design of active topological photonic devices with new or improved functionality.

Probing Edge State Conductance in Ultra‐Thin Topological Insulator Films

AbstractQuantum spin Hall (QSH) insulators have unique electronic properties, comprising a band gap in their 2D interior and 1D spin‐polarized edge states in which current flows ballistically. In scanning tunneling microscopy (STM), the edge states manifest themselves as an enhanced local density of states (LDOS). However, there is a significant research gap between the observation of edge states in nanoscale spectroscopy, and the detection of ballistic transport in edge channels which typically relies on transport experiments with microscale lithographic contacts. Here, few‐layer films of the 3D topological insulator (BixSb)2Te3 are studied, for which a topological transition to a 2D topological QSH insulator phase has been proposed. Indeed, an edge state in the LDOS is observed within the band gap. Yet, in nanoscale transport experiments with a four‐tip STM, two‐quintuple‐layer films do not exhibit a ballistic conductance in the edge channels and thus no QSH edge states. This demonstrates that the detection of edge states in spectroscopy can be misleading with regard to the identification of a QSH phase. In contrast, nanoscale multi‐tip transport experiments are a robust method for effectively pinpointing ballistic edge channels, as opposed to trivial edge states, in quantum materials.

Reliable community search in dynamic networks

Searching for local communities is an important research problem that supports advanced data analysis in various complex networks, such as social networks, collaboration networks, cellular networks, etc. The evolution of such networks over time has motivated several recent studies to identify local communities in dynamic networks. However, these studies only utilize the aggregation of disjoint structural information to measure the quality and ignore the reliability of the communities in a continuous time interval. To fill this research gap, we propose a novel (θ, k )- core reliable community (CRC) model in the weighted dynamic networks, and define the problem of most reliable community search that couples the desirable properties of connection strength, cohesive structure continuity, and the maximal member engagement. To solve this problem, we first develop a novel edge filtering based online CRC search algorithm that can effectively filter out the trivial edge information from the networks while searching for a reliable community. Further, we propose an index structure, Weighted Core Forest-Index (WCF-index), and devise an index-based dynamic programming CRC search algorithm, that can prune a large number of insignificant intermediate results and support efficient query processing. Finally, we conduct extensive experiments systematically to demonstrate the efficiency and effectiveness of our proposed algorithms on eight real datasets under various experimental settings.

Random-Gate-Voltage Induced Al’tshuler–Aronov–Spivak Effect in Topological Edge StatesSupported by the National Natural Science Foundation of China (Grant Nos. 12074172, 11674160, and 11974168), the StartupGrant at Nanjing University, the State Key Program for Basic Researches of China (Grant No. 2017YFA0303203), and the ExcellentProgramme at Nanjing University

Helical edge states are the hallmark of the quantum spin Hall insulator. Recently, several experiments have observed transport signatures contributed by trivial edge states, making it difficult to distinguish between the topologically trivial and nontrivial phases. Here, we show that helical edge states can be identified by the random-gate-voltage induced Φ 0/2-period oscillation of the averaged electron return probability in the interferometer constructed by the edge states. The random gate voltage can highlight the Φ 0/2-period Al’tshuler–Aronov–Spivak oscillation proportional to sin2(2πΦ/Φ 0) by quenching theΦ 0-period Aharonov–Bohm oscillation. It is found that the helical spin texture induced π Berry phase is key to such weak antilocalization behavior with zero return probability at Φ = 0. In contrast, the oscillation for the trivial edge states may exhibit either weak localization or antilocalization depending on the strength of the spin-orbit coupling, which has finite return probability at Φ = 0. Our results provide an effective way for the identification of the helical edge states. The predicted signature is stabilized by the time-reversal symmetry so that it is robust against disorder and does not require any fine adjustment of system.

Zeta functions of edge-free quotients of graphs

We define the Ihara zeta function ζ(u,X) and Artin–Ihara L-function of the quotient graph of groups X=Y//G, where G is a group acting on a finite graph Y with trivial edge stabilizers. We determine the relationship between the primes of Y and X and show that the projection map Y→X can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the L-function of X evaluated at the regular representation is equal to ζ(u,Y), and that ζ(u,X) divides ζ(u,Y). We derive two-term and three-term determinant formulas for the zeta and L-functions, and compute several examples of L-functions of edge-free quotients of the tetrahedron graph K4.

Extreme boundary conditions and random tilings

Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]

Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices

We report on the experimental demonstration of two-photon quantum walks at the edge of a photonic Su–Schrieffer–Heeger lattice and compare them to those observed when launching photons at the edge of a homogeneous lattice. Whereas at the topological edge, one of the photons primarily remains close to the edge, both photons penetrate freely from the trivial edge into the bulk. This behavior manifests also in the average inter-particle distance, which is significantly larger at the topological edge. Hence, for a given propagation length, the entangled two-photon state launched at the topological edge extends over a wider domain of the lattice.

Controlling the Quantum Spin Hall Edge States in Two-Dimensional Transition Metal Dichalcogenides.

Two-dimensional transition metal dichalcogenides (TMDs) of Mo and W in their 1T' crystalline phase host the quantum spin Hall (QSH) insulator phase. We address the electronic properties of the QSH edge states by means of first-principles calculations performed on realistic models of edge terminations of different stoichiometries. The QSH edge states show a tendency to have complex band dispersions and coexist with topologically trivial edge states. We nevertheless identify two stable edge terminations that allow isolation of a pair of helical edge states within the band gap of TMDs, with monolayer 1T'-WSe2 being the most promising material. We also characterize the finite-size effects in the electronic structure of 1T'-WSe2 nanoribbons. Our results provide guidance to the experimental studies and possible practical applications of QSH edge states in monolayer 1T'-TMDs.

Conductance of quantum spin Hall edge states from first principles: The critical role of magnetic impurities and inter-edge scattering

The outstanding transport properties expected at the edge of two-dimensional time-reversal invariant topological insulators have proven to be challenging to realize experimentally, and have so far only been demonstrated in very short devices. In search for an explanation to this puzzling observation, we here report a full first-principles calculation of topologically protected transport at the edge of novel quantum spin Hall insulators - specifically, Bismuth and Antimony halides - based on the non-equilibrium Green's functions formalism. Our calculations unravel two different scattering mechanisms that may affect two-dimensional topological insulators, namely time-reversal symmetry breaking at vacancy defects and inter-edge scattering mediated by multiple co-operating impurities, possibly non-magnetic. We discuss their drastic consequences for typical non-local transport measurements as well as strategies to mitigate their negative impact. Finally, we provide an instructive comparison of the transport properties of topologically protected edge states to those of the trivial edge states in MoS$_2$ ribbons. Although we focus on a few specific cases (in terms of materials and defect types) our results should be representative for the general case and thus have significance beyond the systems studied here.

Noise processes in InAs/Ga(In)Sb Corbino structures

Two-dimensional topological insulators are of great interest, with predicted topological protection of one-dimensional helical edge states at their boundaries. Shot noise, the fluctuations in driven current due to the discreteness of charge carriers, has been proposed as a way of distinguishing between trivial and nontrivial edge state conduction, as well as a means of assessing back-scattering mechanisms in the latter. Such measurements require an understanding of possible contributions to the noise from contacts and conduction in the 2D bulk. We present noise measurements in Corbino structures based on InAs/Ga(In)Sb quantum well interfaces over a broad temperature and applied current range. As the temperature is lowered and the bulk transport is gapped out, shot noise becomes detectable in these two-terminal devices, in both high- and low-frequency measurement techniques. Quantitative comparison with a noise model shows that the total applied voltage drop is split among the contacts and the bulk and that the devices have some intrinsic asymmetry. Within that model, the magnitude of the shot noise appears to be anomalously large, implying the contacts to the 2D bulk are nontrivial in this system.

Localization of trivial edge states in InAs/GaSb composite quantum wells

The InAs/GaSb heterostructure is one of the systems where the quantum spin Hall effect is predicted to arise. However, as confirmed by recent experimental studies, the most significant highlight of the effect, i.e., conductance quantization due to nontrivial edge states, is obscured by spurious conductivity arising from trivial edge states. In this Rapid Communication, we present an experimental observation of the strong localization of trivial edge modes in an InAs/GaSb heterostructure which was weakly disordered by silicon deltalike dopants within the InAs layer. The edge conduction which is characterized by a temperature-independent behavior at low temperatures and a power law at high temperatures is observed to be exponentially scaled with the length of the edge. Comprehensive analyses on measurements with a range of devices is in agreement with the localization theories in quasi-one-dimensional electronic systems.

Competing edge structures of Sb and Bi bilayers generated by trivial and nontrivial band topologies

One-dimensional (1D) edge states formed at the boundaries of 2D normal and topological insulators have shown intriguing quantum phases such as charge density wave and quantum spin Hall effect. Based on first-principles density-functional theory calculations including spin-orbit coupling (SOC), we show that the edge states of zigzag Sb(111) and Bi(111) nanoribbons drastically change the stability of their edge structures. For zigzag Sb(111) nanoribbon, the Peierls-distorted or reconstructed edge structure is stabilized by a band-gap opening. However, for zigzag Bi(111) nanoribbon, such two insulating structures are destabilized due to the presence of topologically protected gapless edge states, resulting in the stabilization of a metallic, shear-distorted edge structure. We also show that the edge states of the Bi(111) nanoribbon exhibit a larger Rashba-type spin splitting at the boundary of Brillouin zone, compared to those of the Sb(111) nanoribbon. Interestingly, the spin textures of edge states in the Peierls-distorted Sb edge structure and the shear-distorted Bi edge structure have all three spin components perpendicular and parallel to the edges, due to their broken mirror-plane symmetry. The present findings demonstrate that the topologically trivial and nontrivial edge states play crucial roles in determining the edge structures of normal and topological insulators.

H/e Superconducting Quantum Interference through Trivial Edge States in InAs.

Josephson junctions defined in strong spin orbit semiconductors are highly interesting for the search for topological systems. However, next to topological edge states that emerge in a sufficient magnetic field, trivial edge states can also occur. We study the trivial edge states with superconducting quantum interference measurements on nontopological InAs Josephson junctions. We observe a SQUID pattern, an indication of superconducting edge transport. Also, a remarkable h/e SQUID signal is observed that, as we find, stems from crossed Andreev states.

Role of helical edge modes in the chiral quantum anomalous Hall state

Although indications are that a single chiral quantum anomalous Hall(QAH) edge mode might have been experimentally detected. There have been very many recent experiments which conjecture that a chiral QAH edge mode always materializes along with a pair of quasi-helical quantum spin Hall (QSH) edge modes. In this work we deal with a substantial ‘What If?’ question- in case the QSH edge modes, from which these QAH edge modes evolve, are not topologically-protected then the QAH edge modes wont be topologically-protected too and thus unfit for use in any applications. Further, as a corollary one can also ask if the topological-protection of QSH edge modes does not carry over during the evolution process to QAH edge modes then again our ‘What if?’ scenario becomes apparent. The ‘how’ of the resolution of this ‘What if?’ conundrum is the main objective of our work. We show in similar set-ups affected by disorder and inelastic scattering, transport via trivial QAH edge mode leads to quantization of Hall resistance and not that via topological QAH edge modes. This perhaps begs a substantial reinterpretation of those experiments which purported to find signatures of chiral(topological) QAH edge modes albeit in conjunction with quasi helical QSH edge modes.