Protected Edge Research Articles

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.

Electronic and spin transport in Bismuthene with magnetic impurities

Topological insulators have remained as candidates for future electronic devices since their first experimental realization in the past decade. The existence of topologically protected edge states could be exploited to generate a robust platform and develop quantum computers. In this work we explore the role of magnetic impurities in the transport properties of topological insulators, in particular, we study the effect on the zigzag bismuthene nanoribbon edge states conductivity. By means of realistic abinitio calculations we simulate the interaction between magnetic adatoms and topological insulators, furthermore, our main goal is to obtain the transport properties for large samples as it would be possible to localize edge states at large scales.

Rigorous analysis of the topologically protected edge states in the quantum spin Hall phase of the armchair ribbon geometry

Studying the edge states of a topological system and extracting their topological properties is of great importance in understanding and characterizing these systems. In this paper, we present a novel analytical approach for obtaining explicit expressions for the edge states in the Kane-Mele model within a ribbon geometry featuring armchair boundaries. Our approach involves a mapping procedure that transforms the system into an extended Su–Schrieffer–Heeger model, specifically a two-leg ladder, in momentum space. Through rigorous derivation, we determine various analytical properties of the edge states, including their wave functions and energy dispersion. Additionally, we investigate the condition for topological transition by solely analyzing the edge states, and we elucidate the underlying reasons for the violation of the bulk-edge correspondence in relatively narrow ribbons. Our findings shed light on the unique characteristics of the edge states in the quantum spin Hall phase of the Kane–Mele model and provide valuable insights into the topological properties of such systems.

Topological Microlaser with a Non-Hermitian Topological Bulk.

Bulk-edge correspondence, with quantized bulk topology leading to protected edge states, is a hallmark of topological states of matter and has been experimentally observed in electronic, atomic, photonic, and many other systems. While bulk-edge correspondence has been extensively studied in Hermitian systems, a non-Hermitian bulk could drastically modify the Hermitian topological band theory due to the interplay between non-Hermiticity and topology, and its effect on bulk-edge correspondence is still an ongoing pursuit. Importantly, including non-Hermicity can significantly expand the horizon of topological states of matter and lead to a plethora of unique properties and device applications, an example of which is a topological laser. However, the bulk topology, and thereby the bulk-edge correspondence, in existing topological edge-mode lasers is not well defined. Here, we propose and experimentally probe topological edge-mode lasing with a well-defined non-Hermitian bulk topology in a one-dimensional (1D) array of coupled ring resonators. By modeling the Hamiltonian with an additional degree of freedom (referred to as synthetic dimension), our 1D structure is equivalent to a 2D non-Hermitian Chern insulator with precise mapping. Our Letter may open a new pathway for probing non-Hermitian topological effects and exploring non-Hermitian topological device applications.

Andreev reflection via the zero-energy bound states at the superconducting skyrmion crystal–normal skyrmion crystal interface

In this work, we used the double exchange model and studied the transport properties of the electronic states across the superconducting skyrmion crystal (SkX)–normal SkX interface. By calculating the energy dispersion and the real-space wave function of the electronic states we found topologically protected edge zero modes in the superconducting SkX nanoribbon. By consideration of the symmetry of the Hamiltonian of the bulk superconducting SkX and its nanoribbon, we found that they belong to the symmetry class D and support Majorana edge modes. To demonstrate the transport properties of these zero edge modes, we used the Green's function method and calculated the Andreev reflection coefficients of the electronic states across the superconducting SkX–normal SkX interface. Zero-mode-induced Andreev reflection peaks are demonstrated and also reflected in the tunneling current. Single-channel conductance is quantized at 2e2/h and 4e2/h for two-fold and four-fold degenerate Majorana bound states, respectively.

Measuring Topological Entanglement Entropy Using Maxwell Relations.

Topological entanglement entropy (TEE) is a key diagnostic of topological order, allowing one to detect the presence of Abelian or non-Abelian anyons. However, there are currently no experimentally feasible protocols to measure TEE in condensed matter systems. Here, we propose a scheme to measure the TEE of chiral topological phases, carrying protected edge states, based on a nontrivial connection with the thermodynamic entropy change occurring in a quantum point contact (QPC) as it pinches off the topological liquid into two. We show how this entropy change can be extracted using Maxwell relations from charge detection of a nearby quantum dot. We demonstrate this explicitly for the Abelian Laughlin states, using an exact solution of the sine-Gordon model describing the universal crossover in the QPC. Our approach might open a new thermodynamic detection scheme of topological states also with non-Abelian statistics.

The Contribution of Women to Peace in the Middle East: The Experience of the Movement Women Wage Peace (WWP)

In the context of the many movements with a strong female presence promoting peace in the Middle East, the movement Women Wage Peace (WWP) currently represents the largest and most impactful organization in the country. Founded in the aftermath of the 50-day Gaza War/Operation Protective Edge of 2014, Women Wage Peace has grown to 45,000 Israeli members, the most significant grassroots peace movement in Israel today. WWP’s theory of change refracts the Israeli–Palestinian conflict, and its resolution, through a gendered lens. The movement is non-partisan and does not support any specific solution to the conflict. Instead, it empowers women from diverse communities to build trust across divides, leading to a unified demand for diplomatic negotiation, with full representation of women, to end the Israeli–Palestinian conflict. WWP enables very diverse women to unite with the aim of taking their own and their children’s futures into their own hands: women from the left, center, and right, young and old, from the center and periphery of the country, religious and secular, Jewish, Arab, Druze, and Bedouin. The movement continues to refine its non-hierarchical structure, distributing its work among thousands of volunteers who serve on regional and/or mission-specific teams, including Government Engagement, Foreign Affairs, Digital Communication, Special Projects, and Strategy.

Bulk-Boundary Correspondence and Singularity-Filling in Long-Range Free-Fermion Chains

The bulk-boundary correspondence relates topologically protected edge modes to bulk topological invariants and is well understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent $\ensuremath{\alpha}$, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with $\ensuremath{\alpha}>1$, linking together the quantized winding invariant, bulk topological string-order parameters, and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalize these results by (i) identifying a family of BDI chains with $\ensuremath{\alpha}<1$ where our results still hold and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when $\ensuremath{\alpha}\ensuremath{-}1$ exceeds the dynamical critical exponent.

Active Control Topological Valley Modes in Metamaterial Plates

Phononic crystals and metamaterials have a unique band structure that allows for the existence of topologically protected surface states. The topologically protected edge states can guide elastic waves without significant scattering or loss of energy. One of the most promising applications of topological insulators in wave guiding is in the field of acoustics, where they can be used to design highly efficient and robust acoustic wave guides. However, the high efficiency, precision, reconfigurability, and robustness of elastic waves remains challenging. The topological insulators provide a feasible method to design high-efficiency, robust, and low-backscattering waveguides. In this work, a novel design of hexagonal metamaterial plates composed of a base plate and piezoelectric patches is proposed. The hexagonal metamaterial plate can generate robust topologically protected edge waves via active control. The paths of the topologically protected edge waves can be tuned by adjusting the control parameters. The robustness and efficiency of the proposed hexagonal metamaterial plate are testified to by the numerical examples. These findings provide systematic theoretical guidelines for designing reconfigurable wave guides, elastic wave splitters, and novel elastic wave devices and hold great promise for the development of high-performance and versatile wave guide technologies with potential applications in a wide range of fields.

Semimetallic spin-density wave state in iron pnictides

We examine the existence of semimetallic spin-density wave (SDW) states in iron pnictides. In the experimentally observed metallic SDW state, the symmetry-protected Dirac cones are located away from the Fermi surface giving rise to tiny pockets and there are also additional Fermi pockets such as one around Γ. We find that the location of a pair of Dirac points with respect to the Fermi surface exhibits significant sensitivity to the orbital splitting between the d xz and d yz orbitals. Besides, in the presence of orbital splitting, the Fermi pockets not associated with the Dirac cones, can be suppressed so that a semimetallic SDW state can be realized. We explain these findings in terms of difference in the slopes and orbital contents of the bands which form the Dirac cone, and obtain the necessary conditions dependent on these two and other parameters for the coexisting Dirac semimetallic and SDW states. Additionally, the topologically protected edge states are studied in the ribbon geometry when the same are oriented either along x or y axes.

What Does Art History Have to Say About a Lebanese Sasquatch? The Body of Decolonial Struggle in Amanda Boulos's Art

Abstract This paper focuses on several works by the Palestinian-Canadian painter Amanda Boulos that communicate the shared desire of both Palestinians in the diaspora and Indigenous peoples of Canada to move beyond the normative identities of settler colonialism. Through co-ordinated social historical, formalist and iconographical readings of Boulos's work, I propose a shift in the discourse on global contemporary art, from postcolonial figures of the oriental, the subaltern and the hybrid to strategies of representation such as transformation, ambiguity and queering – a shift intended to foster alliances amongst members of BIPOC communities, against the divisive politics of settler colonialism in Palestine, Canada and elsewhere. To this end, the artworks are set in two interrelated contexts: one art historical and one more strictly political. Boulos's work is read in relation to that of contemporary artists dealing with postcolonial conflict zones, and Modern and contemporary Palestinian artists both in Palestine and its diaspora. Relevant political contexts for her work include Palestinian LGBTQ activism after the Second Intifada and the 2014 Israeli War on Gaza called Operation Protective Edge.

Tailoring of interface modes in topologically protected edge states with hourglass lattice metamaterials

Nonreciprocity and topologically protected wave propagation have significant implications on how energy and information are transmitted or guided within materials to control or mitigate its effects. The major challenge in tailoring interface mode arises from challenges related to the customizability and linearity of interface lattice, moreover, there is a scarce of experimental analysis reported in the literature. Our study has focused on obtaining topologically protected nontrivial interface modes at a specific frequency by breaking the inversion symmetry through novel hourglass metastructure both theoretically and experimentally. Detailed work on wave transmission, dispersion, and bandgap analysis are carried out considering topological metamaterials. New cellular configurations based on regular honeycomb and auxetic cells, and variations of their geometric parameters responsible for interface mode tuning are reported here. A generalized theoretical scheme for different combinations of the hourglass lattice is derived at the interface, and consequent energy harvesting and damping prospects are reported. Analytical modeling of topological metamaterial lattice along with numerical simulation, additive layer manufacturing (3D printing), and finally experimental validations are carried out to justify the behavior and reveal the underlying physics responsible for its unique behavior. Three types of configurations including hourglass lattice at the interface define a general framework for introducing lattice-based imperfections in the continuous elastic structure for potential engineering applications. The localized topological interface mode obtained within the bandgap can be tuned significantly with the help of latticed hourglass and may be utilized for the purpose of wave guiding, wave focusing, and energy harvesting within the isolation zone.

Topological edge solitons in the non-Hermitian nonlinear Su-Schrieffer-Heeger model

The Su-Schrieffer-Heeger model is the simplest one-dimensional model showing the characteristic features of topological insulators. Its most interesting property is the appearance of a solution of the edge state or edge soliton in a topologically nontrivial phase determined by the system parameters. Recently, the authors have been investigating generalizations of such a system in two different aspects, both through the inclusion of nonlinearity in the model, and considering the effects of gain and loss. This paper provides an example of accounting for both of these mechanisms. It is shown that for a given gain parameter, there is a region of loss parameters where protected edge solitons are also implemented. Passing the critical value of the loss parameter, the stationary edge soliton becomes oscillating. It is interesting that there are regimes in which the edge soliton, while maintaining spatial localization, demonstrates chaotic temporal dynamics. Analytical estimates characterizing the properties of solutions are given.

Robust beam splitter with fast quantum state transfer through a topological interface

The Su–Schrieffer–Heeger (SSH) model, commonly used for robust state transfers through topologically protected edge pumping, has been generalized and exploited to engineer diverse functional quantum devices. Here, we propose to realize a fast topological beam splitter based on a generalized SSH model by accelerating the quantum state transfer (QST) process essentially limited by adiabatic requirements. The scheme involves delicate orchestration of the instantaneous energy spectrum through exponential modulation of nearest neighbor coupling strengths and onsite energies, yielding a significantly accelerated beam splitting process. Due to properties of topological pumping and accelerated QST, the beam splitter exhibits strong robustness against parameter disorders and losses of system. In addition, the model demonstrates good scalability and can be extended to two-dimensional crossed-chain structures to realize a topological router with variable numbers of output ports. Our work provides practical prospects for fast and robust topological QST in feasible quantum devices in large-scale quantum information processing.

Rotational symmetry protected edge and corner states in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with non-trivial strong topological invariants (protected by internal symmetries) with additional "weak" topological indices. In this paper, we study the edge physics of systems with a non-trivial shift invariant, which is protected by either a continuous $\text{U}(1)_r$ or discrete $\text{C}_n$ rotation symmetry, along with internal $\text{U}(1)_c$ charge conservation. Specifically, we construct an interface between two systems which have the same Chern number but are distinguished by their Wen-Zee shift and, through analytic arguments supported by numerics, show that the interface hosts counter-propagating gapless edge modes which cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian sub-algebra framework for gapped boundaries while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and compute them in the case where the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in symmetry enriched topological phases.

Tailored Topological Edge Waves via Chiral Hierarchical Metamaterials

Precise manipulation of the direction and re-direction of vibrational wave energy is a key demand in wave physics and engineering. We consider the paradigm of a finite frame-like structure and the requirement to channel energy away from critical regions, leaving them vibration-free, and redirect energy along edges towards energy concentrators for damping or energy harvesting. We design an exemplar frame metamaterial, combining two distinct areas of wave physics. Firstly, topological edge states taking an unconventional tetrachiral lattice. We control these highly localised protected edge states leveraging a hierarchy of scales through the addition of micro-resonators that impose tuneable symmetry breaking and reconfigurable mass. This allows us to achieve precise positional control in the macro-scale frame lattice, thereby opening opportunities for robust signal transport and vibration control. Experiments, theory, simulation are all utilised to provide a comprehensive analysis and interpretation of the physics.

RKKY interactions mediated by topological states in transition metal doped bismuthene

We have investigated magnetic interactions between transition metal ions in bismuthene topological insulators with protected edge states. We find that these topological states have a crucial role in the magnetic interactions in 2D topological insulators. Using first-principles and model Hamiltonian, we make a comparative study of transition metal doped bulk and nanoribbon bismuthene. While a direct overlap between the transition metal prevails in gapped bulk bismuthene, at the borders of nanoribbons, a long-range magnetism is present. The exchange interactions are well described by a Ruderman–Kittel–Kasuya–Yosida-like Hamiltonian mediated by massive and topological states. Our results show a dominance of antiferromagnetism promoted by the topological states, preserving the spin-locked Dirac crossing states due to a global time-reversal symmetry preservation. This extended magnetic interactions mediated mainly by massless electrons can increase the spin diffusion length being promising for fast dissipationless spintronic devices.

Two-particle topological Thouless spin pump

We show that two particles interacting via spin exchange exhibit topological features found in one-dimensional single particle lattice models. This is accomplished by absorbing all of the spatial degrees of freedom of the lattices into the spin degrees of freedom of the two particles. Comparing the spin system with the Su-Schrieffer-Heeger model, we show the existence of topologically protected edge spin states and establish the bulk-edge correspondence. Modifying the spin system with a chiral symmetry breaking term results in it resembling the Rice-Mele model and can therefore act as a Thouless spin pump of one of the particles when periodically and adiabatically driven. By using the spin states as a synthetic spatial dimension, we show two particles are enough to simulate well known topological properties in condensed matter physics.

One-dimensional noninteracting topological insulators with chiral symmetry

We construct microscopical models of one-dimensional non-interacting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII and CI. We find that the $\mathbb{Z}$ topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operator changes the sign of the winding number. The freedom to choose the sign of the chiral symmetry operator on each chain independently allows us to construct two distinct possible chiral symmetry operators when the chains are weakly coupled -- in one case, the total winding number is given by the sum of the winding number of individual chains while in the second case, the difference is taken. We find that the chiral models that belong to $\mathbb{Z}$ classes, AIII, BDI and CII are topologically equivalent, so they can be adiabatically deformed into one another so long as the chiral symmetry is preserved. We study the properties of the edge states in the constructed models and prove that topologically protected edge states must all be localised on the same sublattice (on any given edge). We also discuss the role of particle-hole symmetry on the protection of edge states and explain how it manages to protect edge states in $\mathbb{Z}_2$ classes, where the integer invariant vanishes and chiral symmetry alone does not protect the edge states anymore. We discuss applications of our results to the case of an arbitrary number of coupled chains, construct possible chiral symmetry operators for the multiple chain case, and briefly discuss the generalisation to any odd number of dimensions.

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.