Higher-order Topological States Research Articles

Higher-order topological corner states and origin in monolayer LaBrO

Abstract Intrinsic higher-order topological insulators driven solely by orbital coupling are rare in electronic materials. Here, we propose that monolayer LaBrO is an intrinsic two-dimensional second-order topological insulator. The generalized second-order topological phase arises from the coupling between the 5d orbital of the La atom and the 2p orbital of the O atom. The underlying physics can be thoroughly described by a four-band generalized higher-order topological model. Notably, the edge states and corner states of monolayer LaBrO exhibit different characteristics in terms of morphology, number, and location distribution under different boundary and nanocluster configurations. Furthermore, the higher-order topological corner states of monolayer LaBrO are robust against variations in spin-orbit coupling and different values of Hubbard U. This provides a material platform for studying intrinsic 2D second-order topological insulators.

Observation of multifunctional robust topological states based on asymmetric C4 photonic crystals

Recent advancements in high-order topological insulators have heralded new opportunities for the innovation and utilization of optical devices. This paper presents a composite asymmetric C4 photonic crystal to achieve multifunctional, robust topological states. Through detailed analysis of the fine changes in the topological bandgap induced by distortion parameters, we facilitate the realization of topological edge states in wavelength division multiplexing applications. We utilize both trivial and nontrivial properties of the topological bandgap to precisely manipulate zero-dimensional angular states, one-dimensional topological boundary states, and two-dimensional body states. Through simulations and experimental results, our advanced asymmetric C4 photonic crystal structure demonstrates superior robustness for the transmission of topological edge states. Our research paves the way for the deployment of more robust topological boundary state transmission systems and advances the application potential of higher-order topological states.

Ideal quadratic nodal point in half-Heusler semimetal LaPtBi

The research focus on topological states in electronic systems has recently expanded from traditional linear dispersions to encompass higher-order dispersions. These higher-order dispersions exhibit unique physical properties, including high topological charge, exotic magnetoresponse, and novel transport behaviors. In this study, we employ first-principles calculations to propose the half-Heusler material LaPtBi as an ideal quadratic nodal point semimetal. Our comprehensive analysis of the band structure reveals an ideal nodal point located precisely at the Fermi level, exhibiting quadratic dispersion in all directions. This finding is supported by symmetry analysis and effective model arguments. Moreover, when the spin-orbit coupling effect is considered, the nodal point transitions from triple to quadruple degeneracy, while maintaining its quadratic dispersion. The material also features multiple prominent arc surface states originating from the nodal point, which are distinctly separated from the bulk bands. This separation facilitates experimental verification. Overall, LaPtBi, with its quadratic nodal point and advantageous properties, serves as an ideal platform for further research. The study of this material is expected to enhance our understanding of higher-order topological states in electronic systems, potentially driving future advancements in the field.

Higher-order topological phase with subsystem symmetries

Abstract A wide variety of higher-order symmetry-protected topological phases (HOSPT) with gapless corners or hinges have been proposed as descendants of topological crystalline insulators protected by spatial symmetry. In this work, we address a new class of higher-order topological states that do not require crystalline symmetries but instead rely on subsystem symmetry for protection. We propose several strongly interacting models with gapless hinges or corners based on a decorated hinge-wall condensate picture. The hinge-wall, which appears as the defect configuration of a Z 2 paramagnet, is decorated with a lower-dimensional SPT state. Such a unique hinge-wall decoration structure leads to gapped surfaces separated by gapless hinges. The non-trivial nature of the hinge modes can be captured by a 1 + 1 D conformal field theory with a Wess–Zumino–Witten term. Moreover, we establish a no-go theorem to demonstrate the ungappable nature of the hinges by making a connection between the generalized Lieb–Schultz–Mattis theorem and the boundary anomaly of the HOSPT state. This universal correspondence engenders a comprehensive criterion to determine the existence of HOSPT under certain symmetries, regardless of the microscopic Hamiltonian.

Vortex superlattice induced second-order topological mid-gap states in first-order topological insulators and superconductors

Topological defects such as vortex and dislocations, support zero-energy localized states as a reflection of the bulk topology, in first-order topological insulators and superconductors. Furthermore, emergent first-order topological mid-gap states have been discovered driven by the magnetic vortex superlattice. However, whether the higher-order topological mid-gap states would emerge from the first-order topological insulators and superconductors with the vortex superlattice remains elusive. In this work, we propose vortex superlattice could induce second-order topological mid-gap states with staggered lattice spacings for vortices in first-order topological insulators and superconductors. These higher-order topological mid-gap states originate from the staggered tunneling between vortex-induced bound states and the emergent π flux on vortex superlattices, as an intrinsic exhibition of the interplay between vortices and bulk topology for the first-order topological states. Our work uncovers higher-topological characteristics of topological-defect superlattice in first-order topological states, and develops a controllable environment for the creation and exploration of higher-order topological states.

Delocalization of higher-order topological states in higher-dimensional non-Hermitian quasicrystals

Delocalization of higher-order topological states in higher-dimensional non-Hermitian quasicrystals

Higher-order topological states in acoustic two-dimensional Penrose quasicrystals

Higher-order topological states in acoustic two-dimensional Penrose quasicrystals

Higher-order topological Dirac phase in Y3InC: a first-principles study

Higher-order topological insulators hosting intriguing topologically protected hinge or corner states are of significant research interest. However, materials that possess higher-order topological hinge states associated with gapless bulk Dirac phases still need to be explored. Using first-principles calculations with hybrid exchange functional, we explore the electronic structure and topological properties of Y3InC and a few of its sister compounds, totaling 16 bulk materials. A symmetry-protected triple point phase, with dominated d-t 2g character, is observed in Y3InC without spin–orbit coupling (SOC). Interestingly, the SOC induces a twin Dirac node phase in the bulk Y3InC. Furthermore, the computed Z 4 topological invariant reveals the higher-order topological nature of investigated materials. To demonstrate the gapless hinge states, we conduct edge state calculations using a rod-shaped geometry of Y3InC. Remarkably, Y3InC is identified to host multi-Dirac nodes in the bulk and surface phases together with the higher-order hinge states. These results lay the groundwork for further experimental and theoretical investigations into cubic antiperovskite materials for higher-order topological phases.

Coupling of photonic topological states and their dynamical control based on liquid crystal.

Optical field manipulation inspired by topology theory has recently drawn great research attention in nanophotonic. For flexible and programmable light management, the capacity to dynamically regulate the photonic topological states in fixed optical artificial microstructures is essential. Here, we propose a dynamic light manipulation of a two-dimensional (2D) photonic lattice aided by liquid crystals, which is composed of all-dielectric photonic crystals with distinct topological phases. In brief, by submerging the well-designed photonic lattice into a liquid crystal (LC), the topological edge and corner states can be actively modulated by applying external bias voltage, which offers an electrically switchable tuning capability, enabling the coupling between higher-order topological states in a structurally deterministic photonic structure. As a proof-of-principle, we use the 1D topological edge states and 0D topological corner states in one sample, respectively, to mimic line-waveguides and corner-cavities, and demonstrate their selective couplings with Fano-like profile driven by electric bias. Our work offers an effective and flexible way for light control in the potential active topological photonic devices.

Higher-order topological states in T-graphene and their realization in photonic crystals

Higher-order topological states extend the power of nontrivial topological states beyond the bulk-edge correspondence. Here we study the higher-order topological states (corner states) in an open-boundary two-dimensional T-graphene lattice. Unlike the common zero-energy corner states, our findings reveal non-zero energy corner states in such lattice systems, and the energy could be controlled by modifying the hopping parameters. Moreover, the corner states could be transferred away from the lattice corners by designing the position-specific vacancy defects. The strong robustness of the corner states is also demonstrated against the uniaxial strain and vacancy defects, respectively. A plasmonic crystal is constructed to testify to the theory, in which the corner states are realized in optical modes and their higher-order topological properties are verified. Our results open the avenue of corner-states engineering, which holds significant physical implications of higher-order topological states for the design of photonic and electronic devices with specialized functionalities.

Creation and manipulation of higher-order topological states by altermagnets

Creation and manipulation of higher-order topological states by altermagnets

Hermitian and non-hermitian higher-order topological states in mechanical metamaterials

Higher-order topological insulators exhibit clear hierarchical boundary states, providing an amazing platform for robust wave manipulations in mechanical or acoustic systems. Recently, this theory has been extended to non-Hermiticity-induced higher-order topology, which enables actively controllable topological transport. Nevertheless, most related studies focus on the non-Hermitian quadrupole topology, associated with the coexistence of negative and positive couplings to constitute a π-flux lattice, which hindered the realizations in classical wave systems. Here, we propose a novel tight-binding model without negative couplings, which supports non-Hermiticity-induced higher-order topological states. In the Hermitian case, the middle band gaps host the vanishing bulk polarization and the associated topology is dependent only upon the associated edge polarization. By introducing the external loss as non-Hermitian components, we reveal that therein the trivial structures can also host topological corner states, featured by the quantized nonzero edge polarizations in the biorthogonal basis. Furthermore, such a topological phase transition induced by non-Hermiticity can be realized for almost all of our lattices, except for the isotropic case where the middle band gaps never open. Basing on the theoretical design, we map the lattice models into the mechanical metamaterials to access the analogous higher-order topology in the elastic wave systems for both Hermitian and non-Hermitian physics. The simulated band structures match well with theoretical solutions. And the robust higher-order topological states are also identified. Our work paves the way to implement the non-Hermitian higher-order topology and offers the possibility for wave manipulations in non-Hermitian systems.

Bogoliubov corner excitations in a conventional s-wave superfluid

Higher-order topological superconductors and superfluids have triggered a great deal of interest in recent years. While Majorana zero-energy corner or hinge states have been studied intensively, whether superconductors and superfluids host higher-order topological Bogoliubov excitations with finite energies remain elusive. In this work, we propose that Bogoliubov corner excitations with finite energies can be induced through only mirror-symmetric local potentials from a trivial conventional s-wave superfluid. The topological Bogoliubov excited modes originate from the nontrivial Bogoliubov excitation bands. These modes are protected by the mirror symmetry and are robust against mirror-symmetric perturbations as long as the Bogoliubov energy gap remains open. Our work provides a new insight into higher-order topological excitation states in superfluids and superconductors.

Acoustic Higher-Order Topological Insulators Induced by Orbital-Interactions.

The discovery of higher-order topological insulator metamaterials, in analogy with their condensed-matter counterparts, has enabled various breakthroughs in photonics, mechanics, and acoustics. A common way of inducing higher-order topological wave phenomena is through pseudo-spins, which mimic the electron spins as a symmetry-breaking degree of freedom. Here, this work exploits degenerate orbitals in acoustic resonant cavities to demonstrate versatile, orbital-selective, higher-order topological corner states. Type-II corner states are theoretically investigated and experimentally demonstrated based on tailored orbital interactions, without the need for long-range hoppings that has so far served as a key ingredient for Type-II corner states in single-orbital systems. Due to the orthogonal nature of the degenerate p orbitals, this work also introduces a universal strategy to realize orbital-dependent edge modes, featuring high-Q edge states identified in bulk bands. These findings provide an understanding of the interplay between acoustic orbitals and topology, shedding light on orbital-related topological wave physics, as well as its applications for acoustic sensing and trapping.

Chern, dipole, and quadrupole topological phases of a simple magneto-optical photonic crystal with a square lattice and an unconventional unit cell

For the study of topological phases in photonics, while quantum-Hall-type first-order chiral edge states are routinely realized in magneto-optical photonic crystals, higher-order topological states are mostly explored in all-dielectric photonic crystals. In this work, we study both first- and second-order topological photonic states in magneto-optical photonic crystals. In specific, we revisit a simple magneto-optical photonic crystal in a square lattice with one gyromagnetic cylinder in each unit cell. However, rather than investigating the conventional unit cell where the cylinder is at the center of the square unit cell as previous works have done, we consider a configuration where the cylinders are located at the four corners of the square unit cell and show that this configuration hosts rich topological phases, such as dual-band Chern, dipole, and quadrupole topological phases. Our detailed characterizations of these topological states are based on the Wannier bands and their polarizations via the Wilson loop and nested Wilson loop methods. We study in detail both the edge and corner states of the different topological phases and show that they exhibit a special feature of “spectrum robustness.” For example, though the edge and corner states of the dipole phases living in a band gap could be pushed into the bulk bands by tuning the boundary conditions, they can pass through the bulk bands and reappear within a different band gap. For dual-band quadrupole phases, we can find a regime where both band gaps host a set of corner states simultaneously and, intriguingly, the filling anomaly of one set of corner states can leave their signatures in the filling anomaly of the other set of corner states though they are separated by an extensive number of bulk states. The rich topological physics revealed in a simple magneto-optical photonic crystal not only provides new insights on higher-order topological phases in time-reversal symmetry-broken photonic systems, the results may also find promising applications by harnessing the potentials of both edge and corner states. Published by the American Physical Society 2024

Higher-order topological states in dual-band valley sonic crystals

As a quantum state of frequency extrema in the momentum space of acoustic systems, sonic valley pseudospin provides a new degree of freedom for controlling acoustic waves. Higher-order topological insulators (HOTIs) have extended the traditional bulk-edge correspondence principle and are a crucial concept for classic wave regulation. However, HOTIs in valley sonic crystals (VSCs) only appear in a single bandgap, which limits the multi-frequency selectivity of the corner state and is not conducive to the design of multi-frequency acoustic communication devices. Here, we demonstrate “Y-shaped” acoustic crystals with C3 symmetry that form a double-band VSC, and the topological phase transitions in both low- and high-frequency band gaps coincide. We realize theoretically and experimentally higher-order states in dual-band valley sonic crystals. Our work enriches the application of HOTIs in acoustic multi-frequency regulatory systems and provides different avenues for designing of multi-band acoustic devices.

Three-dimensional non-Abelian Bloch oscillations and higher-order topological states

Recently, higher-order topological insulators (HOTIs) have been introduced, and were shown to host topological corner states under the theoretical framework of Benalcazar-Bernevig-Hughes. Here we unveil some topological effects in HOTIs by studying the three-dimensional (3D) non-Abelian Bloch oscillations (BOs). In HOTIs, BOs with a multiplied period occur when a force with a special direction is applied due to the effect of the non-Abelian Berry curvature. Along the direction of the oscillations we find a higher-order topological state that goes beyond the theoretical framework of multipole moments. The emergence of such a higher-order topological state coincides with the appearance of the 3D non-Abelian BOs. That is, the 3D non-Abelian BOs can be used as a tool to probe higher-order topological states. These phenomena are observed experimentally with designed electric circuit networks. Our work opens up a way to detect topological phases theoretically and experimentally.

Acoustic higher-order topological states in kagome lattice with split-ring resonators

The next-nearest-neighbour (NNN) hoppings can induce many unique topological phenomena but are rarely considered in acoustics. This work investigates the higher-order topological states induced by the NNN hoppings in acoustic kagome lattice with split-ring resonators (SRRs). We build an extended tight-binding model (TBM) considering NNN hoppings to predict the topological corner modes (CMs) of the acoustic system and obtain the coupling strengths between the SRRs in different rotational angles. During rotating the SRRs, two nontrivial and one trivial systems characterized by bulk polarization can be obtained. In these two nontrivial systems, four novel CMs induced by the NNN hoppings can be observed in the first bandgap. Furthermore, the interactions between resonant and Bragg scattering effect of the SRRs are researched and the defect modes (DMs) are realized in the second bandgap. Finally, the CMs predicted by the extended TBM and DMs match well with the numerical and experimental ones. These discoveries unveil the intricate physics driven by the NNN hoppings, paving the way for innovative approaches to the design of tunable sound energy localization and multi-band metamaterials.

Observation of higher-order topological corner states in photonic two-dimensional trimer lattices.

We demonstrate the first, to the best of our knowledge, experimental observation of higher-order topological corner states in the photonic two-dimensional (2D) trimer lattices. Using a femtosecond laser direct writing technology, we experimentally fabricate a series of 2D trimer lattices with different open boundary conditions and thereby observe two kinds of 0D topological corner states, i.e., topological corner states and topological defect corner states. Interestingly, these corner states and defect corner states can not only exist in the bandgap but also coexist with the bulk states and show obvious localization properties. This work provides fresh perspectives on higher-order topology in artificial microstructures.

Non-Hermitian higher-order topological corner states on the extended kagome lattice

Exploring the interaction between topological phases and non-Hermitian potentials such as gain and loss can benefit designing robust optical devices. Recent studies have revealed topological phases can be simply from gain and loss in non-Hermitian systems. Here, we propose an extended kagome lattice model, where the non-Hermitian potentials drive the system from a trivial phase to a higher-order topological phase. Higher-order topological insulators exhibit lower-dimensional boundary states on corners or hinges. We construct two-dimensional higher-order topological insulators on different arrays of the extended kagome lattice model. Topologically protected states emerge at the corner with a 1/3 fractional charge at each corner as the strength of the gain and loss increases. The topologically protected corner states are characterized by the quantized polarization as the topological index. We find that non-Hermitian potentials provide an extra degree of freedom to switch on and off the higher-order topological corner states. The proposed system can be verified through many experimental platforms, including coupled optical resonating cavities and waveguides. Our work indicates the great prospects for constructing integrated photonics platforms and designing actively reconfigurable photonic devices.