Higher-order topological semimetal in acoustic crystals

Hinge Modes of Surface Arcs in a Synthetic Weyl Phononic Crystal

Chiral bulk Landau levels and surface arcs, as the two distinctive features unique to Weyl semimetals, have each attracted enormous interest. Recent works have revealed that surface-arc modes can support one-sided chiral hinge modes, a hallmark of the three-dimensional quantum Hall effect, as a combined result of chiral Landau levels of bulk states and magnetic response of surface arcs. Here, we exploit a two-dimensional phononic crystal to construct an ideal Weyl semimetal under a pseudomagnetic field, in which a structural parameter is combined to construct a synthetic three-dimensional space. By directly measuring the acoustic pressure fields, we have not only visualized the one-sided chiral hinge modes, but also observed the quantized Landau level modes. The results pave the way to explore the high-dimensional quantum Hall physics in low-dimensional phononic platforms.

Tunable hybrid-order Weyl semimetal via staggered magnetic flux

We investigate a hybrid-order Weyl semimetal (HOWS) constructed by stacking the two-dimensional kagome lattice with staggered magnetic flux. By adjusting the magnitude of flux, higher-order topological phases are tunably intertwined with the first-order topological Chern insulators, which is governed by the evolution of Weyl points. Meanwhile the surface Fermi arcs undergo topological Lifshitz transition. Notably, due to the breaking of time-reversal symmetry (TRS), a novel split of a quadratic double Weyl point occurs, giving rise to additional three type-II Weyl points hybridizing with one type-I node. This phenomenon plays a crucial role in realizing high-Chern-number phases withC=±2and reveals a new mechanism for the emergence of type-II Weyl fermions in topological kagome semimetals. We anticipate that this study will stimulate further investigation into the unique physics of kagome materials and Weyl semimetals.

Topological Network Modes in a Twisted Moiré Phononic Crystal

Twisted moiré materials, a new class of layered structures with different twist angles for neighboring layers, are attracting great attention because of the rich intriguing physical phenomena associated with them. Of particular interest are the topological network modes, first proposed in the small angle twisted bilayer graphene under interlayer bias. Here we report the observations of such topological network modes in twisted moiré phononic crystals without requiring the external bias fields. Acoustic topological network modes that can be constructed in a wide range of twist angles are both observed in the domain walls with and without reconstructions, which serve as the analogy of the lattice relaxations in electronic moiré materials. Topological robustness of the topological network modes is observed by introducing valley-preserved defects to the network channel. Furthermore, the network can be reconfigured into two-dimensional patterns with any desired connectivity, offering a unique prototype platform for acoustic applications.

Demonstration of Acoustic Higher-Order Topological Stiefel-Whitney Semimetal

The higher-order topological phases have attracted intense attention in the past years, which reveals various intriguing topological properties. Meanwhile, the enrichment of group symmetries with projective symmetry algebras redefines the fundamentals of topological matter and makes Stiefel-Whitney (SW) classes in classical wave systems possible. Here, we report the experimental realization of higher-order topological nodal loop semimetal in an acoustic system and obtain the inherent SW topological invariants. In stark contrast to higher-order topological semimetals relating to complex vector bundles, the hinge and surface states in the SW topological phase are protected by two distinctive SW topological charges relevant to real vector bundles. Our findings push forward the studies of SW class topology in classical wave systems, which also show possibilities in robust high-Q-resonance-based sensing and energy harvesting.

Discovery of Higher-Order Nodal Surface Semimetals

The emergent higher-order topological insulators significantly deepen our understanding of topological physics. Recently, the study has been extended to topological semimetals featuring gapless bulk band nodes. To date, higher-order nodal point and line semimetals have been successfully realized in different physical platforms. However, for the conceptually expected higher-order nodal surface semimetals, a concrete model has yet to be proposed, let alone experimentally observed. Here, we report an ingenious design route for constructing this unprecedented higher-order topological phase. The three-dimensional model, layer-stacked with a two-dimensional anisotropic Su-Schrieffer-Heeger lattice, exhibits appealing hinge arcs connecting the projected nodal surfaces. Experimentally, we realize this new topological phase in an acoustic metamaterial, and present unambiguous evidence for both the bulk nodal structure and hinge arc states, the two key manifestations of the higher-order nodal surface semimetal. Our findings can be extended to other classical systems such as photonic, elastic, and electric circuit systems, and open new possibilities for controlling waves.

Pseudospin-dependent acoustic topological edge and corner states in silica aerogel metamaterialsa)

Fueled by the concepts of topological insulators, analogous topological acoustics offer an alternative approach to manipulate sound. Theoretical proposals for subwavelength acoustic topological insulators are considered to be ideal effective parameters or utilizeing artificial coiling-space metamaterials. However, the corresponding realization using realistic soft metamaterials remains challenging. In this study, we present the design of an acoustic subwavelength second-order topological insulator using nanoscale porous solid material, silica aerogel, which supports pseudospin-dependent topological edge and corner states simultaneously. Through simulations and experiments, we demonstrate that silica aerogel can function as a soft acoustic metamaterial at the subwavelength scale. By embedding silica aerogel in an air matrix to construct a honeycomb lattice, a double Dirac cone is obtained. A topological phase transition is induced by expanding or contracting the supercell, resulting in band inversion. Additionally, we propose topologically robust acoustic transmission along the one-dimensional edge. Furthermore, we discover that the proposed sonic crystal sustains zero-dimensional corner states, which can efficiently confine energy at subwavelength corners. These findings offer potential for the realization of subwavelength topological acoustic devices using realistic soft metamaterials.

Higher-order topological phases in crystalline and non-crystalline systems: a review

In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

Observation of D-class topology in an acoustic metamaterial

Topological materials and metamaterials opened new paradigms to create and manipulate phases of matter with unconventional properties. Topological D-class phases (TDPs) are archetypes of the ten-fold classification of topological phases with particle-hole symmetry. In two dimensions, TDPs support propagating topological edge modes that simulate the elusive Majorana elementary particles. Furthermore, a piercing of π-flux Dirac-solenoids in TDPs stabilizes localized Majorana excitations that can be braided for the purpose of topological quantum computation. Such two-dimensional (2D) TDPs have been a focus in the research frontier, but their experimental realizations are still under debate. Here, with a novel design scheme, we realize 2D TDPs in an acoustic crystal by synthesizing both the particle-hole and fermion-like time reversal symmetries for a wide range of frequencies. The design scheme leverages an enriched unit cell structure with real-valued couplings that emulate the targeted Hamiltonian of TDPs with complex hoppings: A technique that could unlock the realization of all topological classes with passive metamaterials. In our experiments, we realize a pair of TDPs with opposite Chern numbers in two independent sectors that are connected by an intrinsic fermion-like time-reversal symmetry built in the system. We measure the acoustic Majorana-like helical edge modes and visualize their robust topological transport, thus revealing the unprecedented D and DIII class topologies with direct evidence. Our study opens up a new pathway for the experimental realization of two fundamental classes of topological phases and may offer new insights in fundamental physics, materials science, and phononic information processing.

Observation of Higher-Order Nodal-Line Semimetal in Phononic Crystals

Higher-order topological insulators and semimetals, which generalize the conventional bulk-boundary correspondence, have attracted extensive research interest. Among them, higher-order Weyl semimetals feature twofold linear crossing points in three-dimensional momentum space, 2D Fermi-arc surface states, and 1D hinge states. Higher-order nodal-point semimetals possessing Weyl points or Dirac points have been implemented. However, higher-order nodal-line or nodal-surface semimetals remain to be further explored in experiments in spite of many previous theoretical efforts. In this work, we realize a second-order nodal-line semimetal in 3D phononic crystals. The bulk nodal lines, 2D drumhead surface states guaranteed by Zak phases, and 1D flat hinge states attributed to k_{z}-dependent quadrupole moments are observed in simulations and experiments. Our findings of nondispersive surface and hinge states may promote applications in acoustic sensing and energy harvesting.

Real higher-order Weyl photonic crystal

Higher-order Weyl semimetals are a family of recently predicted topological phases simultaneously showcasing unconventional properties derived from Weyl points, such as chiral anomaly, and multidimensional topological phenomena originating from higher-order topology. The higher-order Weyl semimetal phases, with their higher-order topology arising from quantized dipole or quadrupole bulk polarizations, have been demonstrated in phononics and circuits. Here, we experimentally discover a class of higher-order Weyl semimetal phase in a three-dimensional photonic crystal (PhC), exhibiting the concurrence of the surface and hinge Fermi arcs from the nonzero Chern number and the nontrivial generalized real Chern number, respectively, coined a real higher-order Weyl PhC. Notably, the projected two-dimensional subsystem with kz = 0 is a real Chern insulator, belonging to the Stiefel-Whitney class with real Bloch wavefunctions, which is distinguished fundamentally from the Chern class with complex Bloch wavefunctions. Our work offers an ideal photonic platform for exploring potential applications and material properties associated with the higher-order Weyl points and the Stiefel-Whitney class of topological phases.

Photonic topological subspace-induced bound states in the continuum

Bound states in the continuum (BICs) are intriguing localized states that possess eigenvalues embedded within the continuum of extended states. Recently, a combination of topological band theory and BIC physics has given rise to a novel form of topological matter known as topological BICs. In this work, we experimentally demonstrate the photonic topological subspace-induced BICs. By using femtosecond-laser writing, we experimentally establish a photonic nontrivial three-leg ladder lattice, thereby directly observe the localized propagation of two kinds of topological edge states which exist at different boundaries. Interestingly, such edge states appear in the continuum of the bulk modes, and the topological properties are inherited from its independent subspace Hamiltonian which contains a celebrated Su-Schrieffer-Heeger lattice. This work not only presents a novel, to the best of our knowledge, platform for investigating topological physics in optics, but also unveils exciting prospects for future exploration of other remarkable BICs.

Stiefel-Whitney topological charges in a three-dimensional acoustic nodal-line crystal

Band topology of materials describes the extent Bloch wavefunctions are twisted in momentum space. Such descriptions rely on a set of topological invariants, generally referred to as topological charges, which form a characteristic class in the mathematical structure of fiber bundles associated with the Bloch wavefunctions. For example, the celebrated Chern number and its variants belong to the Chern class, characterizing topological charges for complex Bloch wavefunctions. Nevertheless, under the space-time inversion symmetry, Bloch wavefunctions can be purely real in the entire momentum space; consequently, their topological classification does not fall into the Chern class, but requires another characteristic class known as the Stiefel-Whitney class. Here, in a three-dimensional acoustic crystal, we demonstrate a topological nodal-line semimetal that is characterized by a doublet of topological charges, the first and second Stiefel-Whitney numbers, simultaneously. Such a doubly charged nodal line gives rise to a doubled bulk-boundary correspondence-while the first Stiefel-Whitney number induces ordinary drumhead states of the nodal line, the second Stiefel-Whitney number supports hinge Fermi arc states at odd inversion-related pairs of hinges. These results experimentally validate the two Stiefel-Whitney topological charges and demonstrate their unique bulk-boundary correspondence in a physical system.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

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 nodal ring photonic semimetal

The intriguing discovery of higher-order topology has tremendously promoted the development of topological physics. Three-dimensional topological semimetals have emerged as an ideal platform for investigating novel topological phases. Consequently, new proposals have been theoretically revealed and experimentally realized. However, most existing schemes are implemented on the acoustic system, while similar concepts are rarely launched in photonic crystals due to the complicated optical manipulation and geometrical design. In this Letter, we propose a higher-order nodal ring semimetal protected by C₂ symmetry originating from C₆ symmetry. The higher-order nodal ring is predicted in three-dimensional momentum space with desired hinge arcs connected by two nodal rings. Fermi arcs and topological hinge modes generate significant marks in higher-order topological semimetals. Our work successfully proves the presence of a novel higher-order topological phase in photonic systems that we will strive to apply practically in high-performance photonic devices.

Square-root higher-order Weyl semimetals

The mathematical foundation of quantum mechanics is built on linear algebra, while the application of nonlinear operators can lead to outstanding discoveries under some circumstances, such as the prediction of positron, a direct outcome of the Dirac equation which stems from the square-root of the Klein-Gordon equation. In this article, we propose a model of square-root higher-order Weyl semimetal (SHOWS) by inheriting features from its parent Hamiltonians. It is found that the SHOWS hosts both “Fermi-arc” surface and hinge states that respectively connect the projection of the Weyl points on the side surface and arris. We theoretically construct and experimentally observe the exotic SHOWS state in three-dimensional (3D) stacked electric circuits with honeycomb-kagome hybridizations and double-helix interlayer couplings. Our results open the door for realizing the square-root topology in 3D solid-state platforms.

The topological properties of square-root Weyl semimetals are derived from the square of the Hamiltonian. Here, the authors propose a tight-binding model for a square-root higher-order Weyl semimetal hosting both Fermi-arc surface and hinge states.

Superconductivity and topological states in hexagonal TaC and NbC

Materials with superconductivity and a nontrivial band structure near the Fermi level are promising candidates in realizing topological superconductivity. Using first-principles calculations, we systematically investigated the stability, mechanical properties, superconductivity, electronic structures, and topological states of hexagonal TaC and NbC. The results show that they are stable and have excellent mechanical properties. We predicted that these two carbides are strong electron-phonon coupling superconductors with superconducting transition temperatures of 14.8 and 17.1 K, respectively. Strong coupling is mainly dominated by in-plane Ta/Nb atomic vibrations and in-plane Ta/Nb-d_xy/d_x²-y² electronic orbitals. The electronic structure calculations demonstrate that a nodal line and a triply degenerate point coexist when not including the spin-orbit coupling (SOC) effect. After including the SOC effect, the nodal line is gapped. The complicated surface states are also calculated and need further experiments to verify. The present results indicate that the hexagonal TaC and NbC are potential candidates as topological superconductors, and pave the way towards exploring the superconductivity and topological materials in condensed matter systems.

Experimental Demonstration of Bulk-Hinge Correspondence in a Three-Dimensional Topological Dirac Acoustic Crystal

Three-dimensional topological Dirac semimetal (DSM) is a vital state to explore topological phases and phase transitions. However, its bulk-boundary correspondence is elusive. Here, we experimentally investigate the higher-order hinge states in an acoustic DSM. Not only removable trivial surface states but also robust nontrivial hinge arcs are observed, attributed to the direct correspondence between bulk polarization and hinge charge. We further reveal that a pair of zigzag and bearded hinges possess arcs located in complementary momentum regions. Our work provides solid proof of the bulk-hinge correspondence in DSM and sheds light on the study of topological hierarchy across dimensions.

Topological dislocation modes in three-dimensional acoustic topological insulators

Dislocations are ubiquitous in three-dimensional solid-state materials. The interplay of such real space topology with the emergent band topology defined in reciprocal space gives rise to gapless helical modes bound to the line defects. This is known as bulk-dislocation correspondence, in contrast to the conventional bulk-boundary correspondence featuring topological states at boundaries. However, to date rare compelling experimental evidences have been presented for this intriguing topological observable in solid-state systems, owing to the huge challenges in creating controllable dislocations and conclusively identifying topological signals. Here, using a three-dimensional acoustic weak topological insulator with precisely controllable dislocations, we report an unambiguous experimental evidence for the long-desired bulk-dislocation correspondence, through directly measuring the gapless dispersion of the one-dimensional topological dislocation modes. Remarkably, as revealed in our further experiments, the pseudospin-locked dislocation modes can be unidirectionally guided in an arbitrarily-shaped dislocation path. The peculiar topological dislocation transport, expected in a variety of classical wave systems, can provide unprecedented control over wave propagations.

Square-root-like higher-order topological states in three-dimensional sonic crystals

The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal (z) direction. With the nontrivial intralayer couplings, the opened degeneracy at theK-Hdirection induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in thezdirection. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.

3D Hinge Transport in Acoustic Higher-Order Topological Insulators

The discovery of topologically protected boundary states in topological insulators opens a new avenue toward exploring novel transport phenomena. The one-way feature of boundary states against disorders and impurities prospects great potential in applications of electronic and classical wave devices. Particularly, for the 3D higher-order topological insulators, it can host hinge states, which allow the energy to transport along the hinge channels. However, the hinge states have only been observed along a single hinge, and a natural question arises: whether the hinge states can exist simultaneously on all the three independent directions of one sample? Here we theoretically predict, numerically simulate, and experimentally observe the hinge states on three different directions of a higher-order topological phononic crystal, and demonstrate their robust one-way transport from hinge to hinge. Therefore, 3D topological hinge transport is successfully achieved. The novel sound transport may serve as the basis for acoustic devices of unconventional functions.

Ideal type-II Weyl points in twisted one-dimensional dielectric photonic crystals

We proposed an one-dimensional layer-stacked photonic crystal using anisotropic materials to realize ideal type-II Weyl points. The topological transition from Dirac to Weyl points can be clearly observed by tuning the twist angle between layers. Also, on the interface between the photonic type-II Weyl material and air, gapless surface states have been demonstrated in an incomplete bulk bandgap. By breaking parameter symmetry, these ideal type-II Weyl points would transform into the non-ideal ones, exhibiting topological surface states with single group velocity. Our work may provide a new idea for the realization of photonic semimetal phases by utilizing naturally anisotropic materials.

Synthetic Three-Dimensional Z×Z_{2} Topological Insulator in an Elastic Metacrystal

We report a three-dimensional (3D) topological insulator (TI) formed by stacking identical layers of Chern insulators in a hybrid real-synthetic space. By introducing staggered interlayer hopping that respects mirror symmetry, the bulk bands possess an additional Z_{2} topological invariant along the stacking dimension, which, together with the nontrivial Chern numbers, endows the system with a Z×Z_{2} topology. A 4-tuple topological index characterizes the system's bulk bands. Consequently, two distinct types of topological surface modes (TSMs) are found localized on different surfaces. Type-I TSMs are gapless and are protected by Chern numbers, whereas type-II gapped TSMs are protected by Z_{2} bulk polarization in the stacking direction. Remarkably, each type-II TSM band is also topologically nontrivial, giving rise to second-order topological hinge modes (THMs). Both types of TSMs and the THMs are experimentally observed in an elastic metacrystal.

Higher-Order Weyl-Exceptional-Ring Semimetals

For first-order topological semimetals, non-Hermitian perturbations can drive the Weyl nodes into Weyl exceptional rings having multiple topological structures and no Hermitian counterparts. Recently, it was discovered that higher-order Weyl semimetals, as a novel class of higher-order topological phases, can uniquely exhibit coexisting surface and hinge Fermi arcs. However, non-Hermitian higher-order topological semimetals have not yet been explored. Here, we identify a new type of topological semimetal, i.e., a higher-order topological semimetal with Weyl exceptional rings. In such a semimetal, these rings are characterized by both a spectral winding number and a Chern number. Moreover, the higher-order Weyl-exceptional-ring semimetal supports both surface and hinge Fermi-arc states, which are bounded by the projection of the Weyl exceptional rings onto the surface and hinge, respectively. Noticeably, the dissipative terms can cause the coupling of two exceptional rings with opposite topological charges, so as to induce topological phase transitions. Our studies open new avenues for exploring novel higher-order topological semimetals in non-Hermitian systems.

Higher-Order Dirac Sonic Crystals

Discovering new topological phases of matter is a major theme in fundamental physics and materials science. Dirac semimetal provides an exceptional platform for exploring topological phase transitions under symmetry breaking. Recent theoretical studies have revealed that a three-dimensional Dirac semimetal can harbor fascinating hinge states, a higher-order topological manifestation not known before. However, its realization in experiment is yet to be achieved. In this Letter, we propose a minimum model to construct a spinless higher-order Dirac semimetal protected by C_{6v} symmetry. By breaking different symmetries, this parent phase transitions into a variety of novel topological phases including higher-order topological insulator, higher-order Weyl semimetal, and higher-order nodal-ring semimetal. Furthermore, for the first time, we experimentally realize this unprecedented higher-order topological phase in a sonic crystal and present an unambiguous observation of the desired hinge states via momentum-space spectroscopy and real-space visualization. Our findings may offer new opportunities to manipulate classical waves such as sound and light.

Transmission properties of microwaves at an optical Weyl point in a three-dimensional chiral photonic crystal

Microwave transmission measurements were performed for a three-dimensional (3D) layer-by-layer chiral photonic crystal (PhC), whose photonic band structure contains 3D singular points, Weyl points. For the frequency and wavevector in the vicinity of a Weyl point, the transmitted intensity was found to be inversely proportional to the square of the propagation length. In addition, the transmitted wave was well-collimated in the plane parallel to the PhC layers, even for point-source incidence. When a plane wave was incident on the PhC containing metal scatters, the planar wavefront was reconstructed after the transmission, indicating a cloaking effect.

Vortex states in an acoustic Weyl crystal with a topological lattice defect

Crystalline materials can host topological lattice defects that are robust against local deformations, and such defects can interact in interesting ways with the topological features of the underlying band structure. We design and implement a three dimensional acoustic Weyl metamaterial hosting robust modes bound to a one-dimensional topological lattice defect. The modes are related to topological features of the bulk bands, and carry nonzero orbital angular momentum locked to the direction of propagation. They span a range of axial wavenumbers defined by the projections of two bulk Weyl points to a one-dimensional subspace, in a manner analogous to the formation of Fermi arc surface states. We use acoustic experiments to probe their dispersion relation, orbital angular momentum locked waveguiding, and ability to emit acoustic vortices into free space. These results point to new possibilities for creating and exploiting topological modes in three-dimensional structures through the interplay between band topology in momentum space and topological lattice defects in real space.

Hybrid-Order Topological Insulators in a Phononic Crystal

Topological phases, including the conventional first-order and higher-order topological insulators and semimetals, have emerged as a thriving topic in the fields of condensed-matter physics and materials science. Usually, a topological insulator is characterized by a fixed order topological invariant and exhibits associated bulk-boundary correspondence. Here, we realize a new type of topological insulator in a bilayer phononic crystal, which hosts simultaneously the first-order and second-order topologies, referred to here as the hybrid-order topological insulator. The one-dimensional gapless helical edge states, and zero-dimensional corner states coexist in the same system. The new hybrid-order topological phase may produce novel applications in topological acoustic devices.