Higher-order Topology Research Articles

Multiple higher-order topological phases with even and odd pairs of zero-energy corner modes in a C3symmetry broken model

Here we study emergent higher-order topological (HOTI) phases in the extended Haldane model without C 3 symmetry. For the inversion symmetric case, the QSHI and QAHI phases can embed the HOTI phases while the remaining QASHI phase does not yield any HOTI phases. Remarkably, four-fold degeneracy of zero-energy corner states can be reduced to two-fold under the application (withdrawal) of sub-lattice mass (Zeeman field) term. The sub-lattice mass and Zeeman field terms compete with each other to pin down the two mid-gap states at zero-energy. Interestingly, the bulk polarization can topologically characterize the second-order topological insulator phase with the mid-gap corner modes irrespective of their energies as long as inversion symmetry is preserved. Our study indicates that a hybrid symmetry can in principle protect the second-order topological insulator phases, however, spin-spectrum gap has to be essentially finite there.

Effective action approach to the filling anomaly in crystalline topological matter

In two dimensions, magnetic higher-order topological insulators (HOTIs) are characterized by excess boundary charge and a compensating bulk ``filling anomaly.'' At the same time, without additional noncrystalline symmetries, the boundaries of two-dimensional HOTIs are gapped and featureless at low energies, while the bulk of the system is predicted to have a topological response to the insertion of lattice (particularly disclination) defects. Until recently, a precise connection between these effects has remained elusive. In this work, we point the direction towards a unifying field-theoretic description for the bulk and boundary response of magnetic HOTIs. By focusing on the low-energy description of the gapped boundary of a two-dimensional magnetic HOTI with no time-reversing symmetries, we show that the boundary charge and filling anomaly arise from the gravitational ``Gromov-Jensen-Abanov'' (GJA) response action [A. Gromov et al., Phys. Rev. Lett. 116, 126802 (2016)] in the context of the quantum Hall effect. As in quantum Hall systems, the GJA action cancels apparent anomalies associated with the bulk response to disclinations, allowing us to derive a concrete connection between the bulk and boundary theories of HOTIs. We show how our results elucidate the connection between higher-order topology and geometric response in band insulators, and point towards a route to understanding interacting higher-order topological phases beyond the simple cases considered here.

Higher-order topological polariton corner state lasing.

Unlike conventional laser, the topological laser is able to emit coherent light robustly against disorders and defects because of its nontrivial band topology. As a promising platform for low-power consumption, exciton polariton topological lasers require no population inversion, a unique property that can be attributed to the part-light-part-matter bosonic nature and strong nonlinearity of exciton polaritons. Recently, the discovery of higher-order topology has shifted the paradigm of topological physics to topological states at boundaries of boundaries, such as corners. However, such topological corner states have never been realized in the exciton polariton system yet. Here, on the basis of an extended two-dimensional Su-Schrieffer-Heeger lattice model, we experimentally demonstrate the topological corner states of perovskite polaritons and achieved polariton corner state lasing with a low threshold (approximately microjoule per square centimeter) at room temperature. The realization of such polariton corner states also provides a mechanism of polariton localization under topological protection, paving the way toward on-chip active polaritonics using higher-order topology.

Role of second-order reputation evaluation in the multi-player snowdrift game on scale-free simplicial complexes

Over the past few decades, the network topology is often characterized via pairwise interactions in the field of indirect reciprocity. However, from human society to ecosystem, interactions usually occur in a group with three or more agents and cannot be fully explained in term of dyadic interactions. In this paper, we improve the multi-player snowdrift game based on scale-free simplicial complexes, and introduce four representative second-order assessment norms to analyze the impact of higher-order topology and reputation evaluation on collective cooperation behaviors. In the model, a focal player i’s general payoff will be regulated by the weight (λ) of player i’s payoff gained in 1-simplices and 2-simplices. Through plenty of Monte Carlo simulations, the results of numerical simulation show that cooperation level can be elevated under certain parameters by introducing non-pairwise interactions. The frequency of cooperation increases (decreases) with the growth of percentage of 2-simplex (ρ) when the cost-to-benefit (r) is smaller (larger). Meanwhile, cooperation behavior can be greatly changed by different reputation evaluation norms (e.g., Shunning rule presents the worst case in terms of the stationary cooperation level, while Image scoring norm creates the highest one). Current results provide some insightful clues for us to comprehend the emergence of mutually beneficial symbiosis in the realistic networked population.

Tunable second-order topological insulators in Chern insulators 2H-FeX2 (X = Cl and Br)

Engineering topological states in two-dimensional (2D) magnets is of pivotal importance to provide significantly rich physics and application potential. Here, we theoretically demonstrate that the second-order topological insulators (SOTIs) with robust nontrivial corner states can be realized in Chern insulators via the widely used strain engineering. The quantum anomalous Hall effect in Chern insulators of honeycomb 2H-FeX2 (X = Cl and Br) is revealed with a nonzero Chern number C=1 and the emergence of metallic chiral edge states. Remarkably, under compressive or tensile strains, topological phase transitions are proposed with the gap-closing in different valleys, giving birth to the 2D SOTIs or trivial insulating 2D magnets. Moreover, large valley polarizations are clearly shown. Our findings open up a promising way for exploring the first- and higher-order topology with intriguing effects.

Acoustic Multiplexing Based on Higher-Order Topological Insulators with Combined Valley and Layer Degrees of Freedom

Higher-order topological insulators (HOTIs) in classical systems, featuring robust multidimensional boundary states protected by various crystalline symmetries, have become a fast-growing research branch in the vast topological family. Therein, valley pseudospins and layer pseudospins are separately introduced as extra degrees of freedom (DOFs) to manipulate topological phases. Here, we experimentally demonstrate that, by combining valley and layer DOFs, intriguing HOTIs can be realized. They host topological edge and corner states that are simultaneously valley dependent and layer polarized. We implement such HOTIs in bilayer sonic crystals (SCs) consisting of carefully stacked triangular scatterers. By rotating the scatterers, the valley and layer DOFs are interplayed, producing combined valley-layer polarizations. Correspondingly, the SCs exhibit versatile sound transport and localization. This inspires us to design an acoustic multiplexer, where sound waves can be guided along specific directions or trapped at specific locations, depending on the excitation frequencies. Our study of higher-order topology based on the interplay between valley and layer DOFs enriches the topological physics. With combined valley and layer polarizations, the resultant HOTIs also offer versatile ways to guide and/or trap sound waves, which have potential applications in integrated and multiplexing phononic devices.

Non-Hermitian higher-order Weyl semimetal with surface diabolic points

Higher-order topology in non-Hermitian (NH) systems has recently become one of the most promising and rapidly developing fields in condensed-matter physics. Many distinct phases that were not present in the Hermitian equivalents are revealed in these systems. In this work, we examine how higher-order Weyl semimetals are impacted by NH perturbation. We identify a new type of topological semimetal, i.e., non-Hermitian higher-order Weyl semimetal (NHHOWS) with surface diabolic points. We demonstrate that in such an NHHOWS, new exceptional points inside the bulk can be created and annihilated, therefore allowing us to manipulate their number. At the boundary, these exceptional points are connected through unique surface states with diabolic points and hinge states. For specific system parameters, the surface of NHHOWS behaves as a Dirac phase with linear dispersion or a Luttinger phase with a quadratic dispersion, thus paving a way for Dirac-Luttinger switching. Finally, we employ the biorthogonal technique to reinstate the standard bulk-boundary correspondence for NH systems and compute the topological invariants. The obtained quantized biorthogonal Chern number and quadruple moment topologically protect the unique surface and hinge states, respectively.

Floquet Weyl semimetal phases in light-irradiated higher-order topological Dirac semimetals

Floquet engineering, the concept of tailoring a system by a periodic drive, is increasingly exploited to design and manipulate topological phases of matter. In this work, we study periodically driven higher-order topological Dirac semimetals associated with a $k$-dependent quantized quadrupole moment by applying circularly polarized light. The undriven Dirac semimetals feature gapless higher-order hinge Fermi arc states which are the consequence of the higher-order topology of the Dirac nodes. Floquet Weyl semimetal phases with hybrid-order topology, characterized by both a $k$-dependent quantized quadrupole moment and a $k$-dependent Chern number, emerge when illumining circularly polarized light. Such Floquet Weyl semimetals support both hinge Fermi arc states and topological surface Fermi arc states. In addition, Floquet Weyl semimetals with tilted Weyl cones in higher-order topological Dirac semimetals are also discussed. Considering numerous higher-order topological Dirac semimetal materials were recently proposed, our findings can be testable soon.

Interaction Effects in a 1D Flat Band at a Topological Crystalline Step Edge.

Step edges of topological crystalline insulators can be viewed as predecessors of higher-order topology, as they embody one-dimensional edge channels embedded in an effective three-dimensional electronic vacuum emanating from the topological crystalline insulator. Using scanning tunneling microscopy and spectroscopy, we investigate the behavior of such edge channels in Pb1-xSnxSe under doping. Once the energy position of the step edge is brought close to the Fermi level, we observe the opening of a correlation gap. The experimental results are rationalized in terms of interaction effects which are enhanced since the electronic density is collapsed to a one-dimensional channel. This constitutes a unique system to study how topology and many-body electronic effects intertwine, which we model theoretically through a Hartree-Fock analysis.

Connecting Higher-Order Topology with the Orbital Hall Effect in Monolayers of Transition Metal Dichalcogenides.

Monolayers of transition metal dichalcogenides (TMDs) in the 2H structural phase have been recently classified as higher-order topological insulators (HOTIs), protected by C_{3} rotation symmetry. In addition, theoretical calculations show an orbital Hall plateau in the insulating gap of TMDs, characterized by an orbital Chern number. We explore the correlation between these two phenomena in TMD monolayers in two structural phases: the noncentrosymmetric 2H and the centrosymmetric 1T. Using density functional theory, we confirm the characteristics of 2H TMDs and reveal that 1T TMDs are identified by a Z_{4} topological invariant. As a result, when cut along appropriate directions, they host conducting edge states, which cross their bulk energy-band gaps and can transport orbital angular momentum. Our linear response calculations thus indicate that the HOTI phase is accompanied by an orbital Hall effect. Using general symmetry arguments, we establish a connection between the two phenomena with potential implications for orbitronics and spin orbitronics.

Acoustic Higher-Order Weyl Semimetal with Bound Hinge States in the Continuum.

Higher-order topological phases have raised widespread interest in recent years with the occurrence of the topological boundary states of dimension two or more less than that of the system bulk. The higher-order topological states have been verified in gapped phases, in a wide variety of systems, such as photonic and acoustic systems, and recently also observed in gapless semimetal phase, such as Weyl and Dirac phases, in systems alike. The higher-order topology is signaled by the hinge states emerging in the common band gaps of the bulk states and the surface states. In this Letter, we report our first prediction and observation of a new type of hinge states, the bound hinge states in the continuum (BHICs) bulk band, in a higher-order Weyl semimetal implemented in phononic crystal. In contrast to the hinge state in gap, which is characterized by the bulk polarization, the BHIC is identified by the nontrivial surface polarization. The finding of the topological BHICs broadens our insight to the topological states, and may stimulate similar researches in other systems such as electronic, photonic, and cold atoms systems. Our Letter may pave the way toward high-Q acoustic devices in application.

Higher-Order Topological Insulator on a Martini Lattice and Its Square Root Descendant

Notion of square-root topological insulators have been recently generalized to higher-order topological insulators. In two-dimensional square-root higher-order topological insulators, emergence of in-gap corner states are inherited from the squared Hamiltonian which hosts higher-order topology. In this paper, we propose that the martini lattice model serves as a concrete example of higher-order topological insulators. Furthermore, we also propose a suquare-root higher-order topological insulator based on the martini model. Specifically, we propose that the honeycomb lattice model with two-site decoration, whose squared Hamiltonian consists of two martini lattice models, realizes square-root higher-order topological insulators. We show, for both of these two models, that in-gap corner states appear at finite energies and that they are portected by non-trivial bulk $\mathbb{Z}_3$ topological invariant.

Analogous quadrupole topology induced by non-Hermiticity

Non-Hermiticity can alter the topological phases of Hermitian lattices and induce gapped edge states and in-gap corner states for the topologically trivial structures in Hermitian cases. To date, however, the realizations of non-Hermiticity-induced higher-order topological phases have often needed intricate designs, where the non-Hermitian systems on a square lattice are comprised of 16 atoms per unit cell with staggered on-site non-Hermitian components at least. It is still necessary to further investigate their topological characteristics in both Hermitian and non-Hermitian systems. In this paper, we suggest an extended quadrupole topological insulator that contains eight atoms in each unit cell and whose topological phases in Hermitian cases are governed by the couplings between adjacent atoms. By introducing external losses into such a higher-order topological insulator, we show that therein the trivial lattices can also give rise to nontrivial quadrupole topology with higher-order corner states. Calculating the biorthogonal nested Wilson loop characterizes the topology for such a non-Hermitian system. Interestingly, the non-Hermitian system has four Wannier bands, with two near zero and two deviating from zero and distributing symmetrically in the negative and positive sections, implying a topology similar to the nontrivial phase in the Hermitian case. Furthermore, both the positive and negative Wannier bands exhibit a nontrivial polarization of 0.5 whereas the remain Wannier bands near zero carry a trivial polarization, indicating that the non-Hermitian system hosts a nontrivial quadrupole topology. We also propose a complete design with acoustic resonators to implement the homologous non-Hermitian topology in acoustic systems to validate our theory. The numerical results match the analytical solution perfectly. Our work introduces an alternative method for fabricating non-Hermitian-induced quadrupole topological insulators and paves the way for future research into the non-Hermitian higher-order topology.

Higher-order topological insulators in hyperbolic lattices

To explore the non-Euclidean generalization of higher-order topological phenomena, we construct a higher-order topological insulator model in hyperbolic lattices by breaking the time-reversal symmetry (TRS) of quantum spin Hall insulators. We investigate three kinds of hyperbolic lattices, i.e., hyperbolic $\{4,5\}$, $\{8,3\}$ and $\{12,3\}$ lattices, respectively. The non-Euclidean higher-order topological behavior is characterized by zero-energy effective corner states appearing in hyperbolic lattices. By adjusting the variation period of the TRS breaking term, we obtain 4, 8 and 12 zero-energy effective corner states in these three different hyperbolic lattices, respectively. It is found that the number of zero-energy effective corner states of a hyperbolic lattice depends on the variation period of the TRS breaking term. The real-space quadrupole moment is employed to characterize the higher-order topology of the hyperbolic lattice with four zero-energy effective corner states. Via symmetry analysis, it is confirmed that the hyperbolic zero-energy effective corner states are protected by the particle-hole symmetry $P$, the effective chiral symmetry $Sm_{z}$, and combined symmetries $C_{p}T$ and $C_{p}m_{z}$. The hyperbolic zero-energy effective corner states remain stable unless these four symmetries are broken simultaneously. The topological nature of hyperbolic zero-energy effective corner states is further confirmed by checking the robustness of the zero-energy modes in the hyperbolic lattices in the presence of disorder. Our paper provides a route for research on hyperbolic higher-order topological insulators in non-Euclidean geometric systems.

Numerical and experimental investigation of second-order mechanical topological insulators

Recently, higher-order topological insulators (HOTIs) as a novel frontier of topological phases of matter have been induced in mechanical systems, opening new routes to manipulate the propagation of elastic waves. Here, second-order mechanical topological insulators (SMTIs) implemented by mechanical metamaterials are systematically investigated in the rectangular lattice, the kagome lattice, the square lattice and the hexagonal lattice. The mechanical metamaterials are constructed from the generalized 2D Su–Schrieffer–Heeger (SSH) models. The topological mechanical metamaterials are characterized by the theories of topological indices and Wannier centers. With simulations and experiments, the corner states and edge states are observed in the topological mechanical metamaterials. Interestingly, the numbers of corner, edge and bulk states are respectively equal to the number of sites located at the corners, edges and bulk. This work offers an inspiring and unified model to study the higher-order topology in mechanical systems, and provides a new way for designing functional and integrated topological devices.

Acoustic graphyne: A second-order real Chern topological insulator

Graphyne has recently attracted much attention since it is an important derivative of graphene with unique topological properties. Although graphyne is not a conventional topological insulator because of its weak spin–orbit coupling, it is a real Chern topological insulator with the higher-order topology. However, it lacks a realistic model. Here, we propose a schedule to realize acoustic graphyne. By introducing negative coupling to simulate the carbon–carbon triple bond, we realize the transition from trivial to higher-order topological phases, characterized by real Chern numbers. These topologically protected corner states are achieved in a finite-size sample, and the condition for their existence is discussed. Our research extends the concept of real Chern insulators and provides a platform for studying the topological properties of graphene-like structural compounds.

Higher-Order Topological States in Thermal Diffusion.

Unlike conventional topological materials that carry topological states at their boundaries, higher-order topological materials are able to support topological states at boundaries of boundaries, such as corners and hinges. While band topology has been recently extended into thermal diffusion for thermal metamaterials, its realization is limited to a 1D thermal lattice, lacking access to the higher-order topology. In this work, the experimental realization is reported of a higher-order thermal topological insulator in a generalized 2D diffusion lattice. The topological corner states for thermal diffusion are observed in the bandgap of diffusion rate of the bulk, as a consequence of the anti-Hermitian nature of the diffusion Hamiltonian. The topological protection of these thermal corner states is demonstrated with the stability of their diffusion profile in the presence of amorphous deformation. This work constitutes the first realization of higher-order topology in purely diffusive systems and opens the door for future thermal management with topological protection beyond 1D geometries.

Square-Root Higher-Order Topology in Rectangular-Lattice Acoustic Metamaterials

Square-root topology offers a distinctive scheme towards topological phases with unconventional spectral properties. In many cases, the formation of square-root topology is governed by the inherent lattice symmetry, with honeycomb lattices being prominent examples. Here, we report on the experimental discovery of square-root higher-order topological insulator phases in an acoustic metamaterial with decorated rectangular lattice. Through acoustic pump-probe techniques, we explore the boundary-dependent topological edge and corner states in both the parent and square-root acoustic metamaterials. We further validate their higher-order topologies as well as their spectral connections via various simulations. Our work not only substantiates the square-root higher-order topology in rectangular-lattice systems from the experimental aspect, but also serves as a step towards versatile higher-order topological phenomena in photonic, acoustic, and mechanical metamaterials.

Seeding and Emergence of Composite Skyrmions in a van der Waals Magnet.

Topological charge plays a significant role in a range of physical systems. In particular, observations of real-space topological objects in magnetic materials have been largely limited to skyrmions - states with a unitary topological charge. Recently, more exotic states with varying topology, such as antiskyrmions, merons, or bimerons and 3D states such as skyrmion strings, chiral bobbers, and hopfions, have been experimentally reported. Along these lines, the realization of states with higher-order topology has the potential to open new avenues of research in topological magnetism and its spintronic applications. Here, real-space imaging of such spin textures, including skyrmion, skyrmionium, skyrmion bag, and skyrmion sack states, observed in exfoliated flakes of the van der Waals magnet Fe3-x GeTe2 (FGT) is reported. These composite skyrmions may emerge from seeded, loop-like states condensed into the stripe domain structure, demonstrating the possibility to realize spin textures with arbitrary integer topological charge within exfoliated flakes of 2D magnets. The general nature of the formation mechanism motivates the search for composite skyrmion states in both well-known and new magnetic materials, which may yet reveal an even richer spectrum of higher-order topologicalobjects.

Ferroelectric higher-order topological insulator in two dimensions

The interplay between ferroelectricity and band topology can give rise to a wide range of both fundamental and applied research. Here, we map out the emergence of nontrivial corner states in two-dimensional ferroelectrics and, remarkably, demonstrate that ferroelectricity and corner states are coupled together by crystallographic symmetry to realize the electric control of higher-order topology. Implementing density functional theory, we identify a series of experimentally synthesized two-dimensional ferroelectrics, such as ${\mathrm{In}}_{2}{\mathrm{Se}}_{3}$, BN, and SnS, as realistic material candidates for the proposed ferroelectric higher-order topological insulators. Our work not only sheds light on traditional ferroelectric materials but also opens an avenue to bridge the higher-order topology and ferroelectricity that provides a nonvolatile handle to manipulate the topology in next-generation electronic devices.