Various topological phases and their abnormal effects of topological acoustic metamaterials

The last 20 years have witnessed growing impacts of the topological concept on the branches of physics, including materials, electronics, photonics, and acoustics. Topology describes objects with some global invariant property under continuous deformation, which in mathematics could date back to the 17th century and mature in the 20th century. In physics, it successfully underpinned the physics of the Quantum Hall effect in 1984. To date, topology has been extensively applied to describe topological phases in acoustic metamaterials. As artificial structures, acoustic metamaterials could be well theoretically analyzed, on‐demand designed, and easily fabricated by modern techniques, such as three‐dimensional printing. Some new theoretical topological models were first discovered in acoustic metamaterials analogous to electronic counterparts, associated with novel effects for acoustics closer to applications. In this review, we focused on the concept of topology and its realization in airborne acoustic crystals, solid elastic phononic crystals, and surface acoustic wave systems. We also introduced emerging concepts of non‐Hermitian, higher‐order, and Floquet topological insulators in acoustics. It has been shown that the topology theory has such a powerful generality that among the disciplines from electron to photon and phonon, from electronic to photonics and acoustics, from acoustic topological theory to acoustic devices, could interact and be analogous to fertilize fantastic new ideas and prototype devices, which might find applications in acoustic engineering and noise‐vibration control engineering in the near future.


| INTRODUCTION
Acoustic metamaterials are artificially designed materials that can exhibit abnormal properties not found in natural materials and offer unconventional ways to control acoustic waves. [1][2][3] Novel properties and unprecedented functionalities arise from the unique microstructures of acoustic metamaterials. The study of acoustic metamaterials can be traced to the emergence of phononic crystals, [4][5][6] inspired by the quantum mechanical band theory of solids and the concept of photonic crystals. [7,8] In the early 1990s, the phononic crystal was first proposed to describe the propagation of acoustic and elastic waves inside periodic structures with the bulk modulus and mass density modulated on the wavelength scale. [9,10] The dispersions of acoustic and elastic waves exhibit energy band structures due to the Bragg scattering effect, and wave propagation is prohibited inside the bandgaps. In the early stage, the research of phononic crystals focused on the formation of bandgaps and related applications. [11,12] Subsequently, unique properties of propagating waves associated with the dispersion of passing bands have been widely studied, such as negative refraction, [13] the zero-refractive index, [14] and slow-wave propagation. In 2000, the concept of local resonance was introduced to design acoustic metamaterials with deepsubwavelength unit cells. [15] The acoustic metamaterials with resonant units can be considered as homogeneous materials with unusually effective parameters. [16] In recent years, significant advances have been achieved in this area, such as transformation acoustics and cloaking, [17] broadband sound absorption, [18] subwavelength imaging, [19] acoustic metasurfaces, [20] and nonreciprocal wave propagation, [21] to name just a few. Among these, one of the latest and most important developments is topological acoustics, inspired by the topological phases in electronic systems.
Topology is a mathematical concept concerned with the properties of objects that are invariant under continuous deformations and stretchings. Over the past century, topology also plays an essential role in physics. In the 1980s, the discovery of the integer quantum Hall effect (QHE) triggered the explosion of investigation of topological phases in condensed-matter physics, [22] where topology describes the global behavior of electronic wavefunctions in momentum space and is robust against local perturbations. The most representative characteristic of topological phases of matter is that they are insulating in the interior but can support backscattering-immune conducting states on their edges. The existence of the unidirectional edge states is required by the bulk-boundary correspondence, which relates the edge states to the bulk topological invariants. After the QHE, tremendous progress has been achieved in this fast-developing field, including the anomalous QHE, the quantum spin Hall effect (QSHE), three-dimensional (3D) topological insulators (TIs), and topological semimetals (TSMs). These exotic phases of matter with robust properties hold promise for future applications in electronic devices.
Over the last decade, motivated by the advances in condensed-matter systems, the study of topological phases has transferred to classical-wave systems, including photonics and acoustics, [23][24][25][26] to realize nontrivial band structures and topologically protected transport for electromagnetic and acoustic waves. Due to the advantages in material design, sample fabrication, and experimental measurement, acoustic wave systems can serve as an excellent and convenient platform for exploring topological physics, which in turn leads to robust designs and novel functionalities for acoustic devices. The acoustic wave systems also have the following additional advantages. There is no Fermi level for acoustic waves, and therefore the acoustic states can be excited and measured at any frequency, which allows the full characterization of the whole band structure. The acoustic "atoms" and couplings among them can be deliberately designed; thus, the "atoms" can be stacked to construct topological phases at different dimensions easily. Moreover, the study of topology phases can be combined with other research fields in acoustic systems, such as non-Hermitian physics. There are also some challenges to overcome for the realization of acoustic topological phases due to the fundamental differences between electrons and phonons. For example, in the QHE, the magnetic field is required to break the time-reversal symmetry (TRS) of the system. The acoustic waves cannot directly interact with the magnetic field; instead, the breaking of TRS can be achieved by applying a circulating airflow, [27] which induces an effective magnetic field for acoustic waves. The acoustic waves also lack the intrinsic spin degree of freedom (DOF). The construction of pseudospin states and Kramers pairs is crucial to emulate the QSHE and other spin-related topological phases in acoustic systems. The realization of topological phases in acoustic systems will strengthen our ability to manipulate acoustic waves and pave the way for various promising applications, such as topological waveguiding, energy harvesting, high-quality acoustic resonators, and onchip phononic circuits.
The structure of this review article is as follows: We first introduce some basic concepts related to topological physics. We give a general overview of the acoustic counterparts of two-dimensional (2D) Quantum Hall systems, 3D TIs, and TSMs. We then discuss the unconventional higher-order topological phases that have gapless boundary states at lower dimensions and non-Hermitian topological phases. We further discuss the dynamic evolution of topological phases, which is related to the topic of Floquet topological phases and topological pumping (TP). We also describe the topological phases in elastic and mechanical metamaterials that hold promise for applications in integrated and solidstate devices. Finally, we will give a brief summary and outlook of topological phases in acoustic metamaterials.

| Famous examples in topology
Topology is a mathematical concept used to investigate continuity and related phenomena, where math is introduced to distinguish objects. As shown in Figure 1, intuitively, many differences between cylinder strip, Mobius strip, ball, and torus can be identified without specific terminology. For instance, we cannot deform a cylinder strip to a Mobius strip without cutting and reconnecting it. The Mobius strip has only one side, and it is a nonorientable surface: if an asymmetric 2D object slides around the strip once, it returns to its starting position as its mirror image. We can further notice that both strips have edges while the ball and torus are closed. A more conspicuous difference between a ball and a torus is the appearance of holes. The hole number difference is captured by a formula: where K is a Gaussian curvature, χ M is an even integer, and g is the number of holes. χ M is unchanged no matter how the structure is continuously deformed. It is a typical topological invariant to characterize the objects where the global property is contracted from local information.
Another example is the well-known hairy ball theorem. The hairy ball theorem states any continuous tangent vector field on the 2-sphere S 2 must have a point where the vector is zero, while nonzero tangent vector fields can exist on the torus T 2 , as shown in Figure 1B, where the vector field is a continuous function of two parameters that parameterize the manifold. The mathematical objects hiding behind are the fiber bundles. A fiber bundle is a structure E B π F ( , , , ) composed of base space B, at each point of which attaches a fiber F and total space E. The map π E B :  a continuous surjection. When a fiber bundle is simply the direct product of a base space and a fiber E B F = × , it is said to be topologically trivial; otherwise, it is nontrivial.

| Bloch band theory
The band structure is closely related to fiber bundles. To show this, we briefly review the Bloch band theory. The lattice is invariant under translation of lattice vectors, from Bloch's theorem, the eigenstates take the form The band is defined in reciprocal lattice with additional freedom k r k r G r = + · · · with G r = mπ · 2 , a direct consequence of translation symmetry. The reciprocal space is the Brillouin zone (BZ), also a base space B which is a direct product of circle S 1 due to its periodic boundary conditions (PBCs) along each dimension: B T = N . Thus, a 2D BZ is also a closed surface with the same topology as a torus T S S = × 2 1 1 , as shown in Figure 1C. The eigenvector r u ( ) k n ( ) can be regarded as the fiber F attached at the base space up to a gauge freedom e k iϕ ( ) . The essential question is how to classify the total space E composed of different fiber F at the same base space, namely, to say they are topologically distinct.

| Berry phase
The Berry phase is a universal concept that emerges in the adiabatic cyclic evolution of states, and it was a generalization of geometric phases, which was first proposed in 1956 to describe the propagation of light through a sequence of polarizers. Berry shows the state evolution will accumulate a phase in addition to the dynamic phase. Under adiabatic cyclic evolution in parameter space, there will be an adiabatic phase associated with it. Many phenomena in classical systems can be attributed to this additional phase. The closed parameter space can be a real space composed of geometric parameters [28] or reciprocal momenta space k.
In band theory, we consider the Bloch band r u ( ) The Berry phase is gauge independent. The integrand of Equation (2) is Berry connection, when a phase freedom k α ( ) is added to the wavefunction. In the crystal background, the Berry phase is also named the Zak phase. The Zak phase of band n gives the expectation value of the position of the electron in the Wannier state coming from band n centered on an atom at the origin. In phononic crystals, none zero Zak phase contributes to wave localization at edges, which was discussed in depth in Xiao et al., [29] in cooperation with traditional surface impedance approaches. None zero Zak phase emerges in various onedimensional (1D) systems, such as a granular chain of periodic quartz cylinders, [30] an elastic beam, [31] and a nanophononic system. [32] Generalizing the Zak phase to a 2D system is natural by defining along a 1D band with a fixed wavevector component along another direction. [33] The Berry phase can be matrix valued when degenerate bands are taken into consideration. The effect of the Berry phase becomes a U J ( ) gauge being the elements of the Berry connection matrix A. Such a matrix-valued Berry phase has already been observed through the holonomic parallel transport of the degenerate modes on a J -sphere, [34] as shown in Figure 2B.

| Berry curvature and Chern number
By taking the curl of the Berry connection, it is possible to obtain a gauge-invariant quantity, where the gradients disappear. The result is the so-called Berry curvature: . If the closed loop C can be viewed as the boundary of a 2D surface S, Equation (1) can be rewritten according to Stocks' theorem.
The integrals of these geometric quantities over a closed manifold-reciprocal momenta space k-will give rise to topological properties and boundary effects, and will be discussed in the following sections. The integration is quantized in 2D energy bands without requiring any underlying symmetry and regardless of the gauge freedom. Such a quantized number is the Chern number, a topological invariant in 2D. Just as the Gauss curvature through a closed surface to be quantized and give the genus, also similar to the magnetic flux through a closed surface to characterize the number of magnetic monopoles. A table to summarize the analogy between band theory and electromagnetism is shown in Table 1. A band with a nonzero Chern number is called topological nontrivial. We can define the gap Chern numbers for a specific bandgap by adding all the band Chern numbers below that bandgap. An insulator with a nonzero gap Chern number is a Chern insulator, in which a celebrated example is the QHE.
The Chern number vanishes for systems with TRS, for which we have k k F F ( ) = − (− ), the integration vanishes. Thus, the Chern insulator requires TRS breaking, its realization in acoustics will be elaborated in Section 3.

| Dirac equation and edge state
A hallmark of a TI is its unusual boundary response, that is, the boundary supports robust boundary states protected by nontrivial bulk topology. In the following sections, we will review various topological systems that support diverse boundary states. As an introductory note, here we introduce Dirac equations widely realized in band theory and show the emergence of boundary states based on the solution of the Dirac equation. A detailed monograph discussing the association between TIs and the solution of Dirac equations can be found in Shen. [35] The Dirac equation describes a spin-1/2 particle with Hamiltonian H c α mc β p = · + 2 , where α i and β are Dirac matrices satisfying the anticommutation relation. In one dimension, the two Dirac matrices are any of two Pauli matrices, considering a lattice background, it can be written as Assuming two media are located at x < 0 and x > 0 such that m x m ( < 0) = − 1 and m x m ( > 0) = + 2 . If we look for a solution with the form ψ x e ( )~i kx , imposing the Dirichlet boundary conditions so that the wavefunction vanishes at infinity, one zero energy solution can be found with the form ( )

Band theory Electromagnetism
The solution locates near x = 0 dominantly and decays exponentially away from the origin, as schematically shown in Figure 3.
It is also possible to find a general zero energy solution for a mass distribution changing from negative to positive at two ends. The solution is quite robust against mass distribution. The solution of the case m m = 1 2 is called the Jackiw-Rebbi solution, which plays an essential role in understanding TIs.

| ACOUSTIC FAMILY OF 2D QHE, 3D TIs, AND TSMs
In 1980, Klitzing discovered the integer QHE, [22] and was later explained by Thouless-Kohmoto-Nightingale-Nijs theory via topological invariant. [36] Since then, the topology-induced quantum or quantum-like effect has been one of the hottest topics not only in condensedmatter physics but also in photonics [24] and acoustics. [25] In recent years, the artificial classical-wave platforms enriched sometimes even advanced the study of these symmetry-protected topological states, including QHE, quantum valley Hall effect (QVHE), [37] QSHE, [8,39] TI, [40] and TSM. [41] A breakthrough is the photonic QHE in 2D magneto-optical crystal under external magnetic bias proposed by Haldane and Raghu in 2005-2008, [42] and its experimental demonstration in 2009. [43] In this section, we attempt to give a brief introduction to acoustic counterparts of the above topological phenomena.

| Dirac physics in 2D acoustic models
Here, we start with 2D Dirac phononic crystals, for example, triangular or honeycomb lattices, where two bulk bands linearly touch at the Dirac point (DP). The C 6v (or C 3v ) group has 2D E 1 /E 2 (or E) representations according to group theory. These lattice symmetries associated with TRS can guarantee linear degenerated nodes at K or K′ point in the first BZ. Generally, the Dirac equation can be described as where ψ 1,2 are the amplitudes of two Bloch states degenerated at DPs. Dirac frequency (ω D ) and velocity (v D ) depend on the structure. In relativistic quantum mechanics, the unusual massless behaviors of DP have attracted intensive attention since the very beginning, such as Zitterbewegung (ZB) [44] (or saying trembling motion) and Klein tunneling. [45] However, they are difficult to directly measure in electronic systems due to their delicate fabrication, complex interaction, and limited scale.
In the past decades, artificial microstructured classicalwave systems, such as photonic and phononic crystals and metamaterials, have shown their ability to provide promising platforms to explore unusual physics, especially in band theory. Particularly for Dirac physics in acoustics, the DP can be constructed in an acoustic triangle lattice for underwater sound, as shown in Figure 4A. [46] Compared with an isolated band whose dispersion is perpendicular to the BZ boundary, this linear node makes its propagating direction and phase elusive. A tiny dip in the transmission spectrum can be found. The transmission around the DP follows the relation as T L 1/  ( Figure 4B). Such extremal transmission, dubbed diffusion transport at DP, differs from either the stopping band case (exponential decay with increasing L) or the passing band case (a constant independent on L). Considering the dynamic behavior near DP, the beating effect can be observed by injecting a Gaussian acoustic pulse with filter frequency widths of 0.01, 0.02, and 0.03 MHz ( Figure 4C). Wider frequency leads to more beating periods. Such a phenomenon originating from the interference between two linear modes near the DP can be deemed as an acoustic analog of ZB. Using Dirac phononic crystals to construct a barrier structure in a sandwich configuration, one further can directly observe Klein tunneling. [47] As shown in Figure 4D, the measured normal transmissions are independent of potential barrier width and height. Moreover, such DP also holds the phase reconstruction effect to realize an acoustic collimator and cloak. [48] On the other hand, bulk-boundary correspondence (BBC) plays a crucial role in topological physics. [49] The topological character of a system is usually exhibited in the lower dimensions than the bulk. For 2D Dirac systems, besides bulk phenomena, the twofold degenerated DPs could host flat edge states when projected on 1D boundaries. [50] Figure 5A,B shows the first BZ with two inequivalent DPs at K or K′ point in a honeycomb lattice. There are three typical boundary shapes, that is, armchair boundary (along the y-direction), a pair of zigzag and bearded boundaries (along the x-direction). For the x-direction case, two DPs are projected on the opposite side of the projected BZ. Figure 5C is the projected band structure. We can find 1D edge arcs (states) related to two DPs for paired zigzag and bearded boundaries appearing in complementary momentum regions. The nontrivial edge arcs are attributed to the correspondence between 2D bulk polarization and 1D boundary charge, which can be analyzed via the Berry phase (quantized to be π or 0) of different unit cell configurations ( Figure 5D). One can imagine how these hierarchal arcs change if the DPs are broken to open a bulk bandgap. Intuitively, there are two possible kinds: (1) upper K′ connecting to upper K̃(or lower to lower); (2) lower K′ connecting to upper K̃(or upper to lower). Figure 6 is a schematic of the acoustic family of 2D QHE based on Dirac phononic crystals. To break twofold F I G U R E 4 Dirac physics in 2D acoustic models. (A) Band structure of a triangle lattice (steel cylinders in water with r/a = 1/3), and transmission along the ГK-direction. The solid (dashed) line corresponds to the thickness of 10a (20a). (B) Diffusion transport at DP (dark dots). The product of transmission (T) and thickness (L) is a constant. The squares (triangles) correspond to another frequency at 0.535 (0.45) MHz. (C) Acoustic ZB under a Gaussian pulse incidence with the center frequency chosen at DP (0.55 MHz). Colors dark, red, and green correspond to 14a, 19a, and 24a thickness, respectively. (A)-(C) are reproduced with permission. [46] Copyright 2008, American Physical Society. (D) Klein tunneling in Dirac phononic crystals with potential barrier of width D and height V. The red and blue lines denote opposite pseudospins. Measured and simulated transmission rate in various cases (bottom panel). Reproduced with permission. [47] Copyright 2020, The American Association for the Advancement of Science. ZB, Zitterbewegung. degenerated bulk DP of two acoustic bands, one may resort to either time-reversal (T) symmetry or spatialinversion (parity, P) symmetry. Thus, the acoustic QHE or QVHE can be expected. Further increasing the DOF to construct fourfold degenerated node with artificial spins for intrinsic spinless longitudinal acoustic waves, layer QVHE and QSHE are also possible via introducing layer DOF or spin-orbit coupling. Their Berry curvatures are plotted in the lower panels. The integration of Berry curvature over the whole BZ is the well-known Berry phase. [51] Berry phase (or saying geometric phase), emerging in the adiabatic cyclic evolution of a state in artificially periodic lattices, is an essential concept to understand these topological phenomena. For the nth isolated band, the periodic part of the acoustic Bloch state denotes as ψ kn , where k represents the Bloch wave vector. The Berry curvature can be defined by

| Acoustic family of 2D QHE
Thus, the corresponding Chern number The Berry curvatures satisfy when P symmetry is kept. That means QHE with nontrivial Chern number must require broken T symmetry. For example, both K and K′ points will acquire the same positive (or negative) Berry curvatures in the QHE (C 1 = ± ) while they are of opposite signs in the valley case (C 0 = ). Near each K or K′ point, we can obtain approximately π phase, corresponding to valley Chern number C = ± v 1 2 . Thus, at a domain wall with opposite C v at each side, their difference becomes for realizing QVHE. Here, the positive (negative) sign refers to clockwise (anticlockwise) chiral edge states in QHE, and -shape (V-shape) dispersion in two distinct types of domain walls in QVHE.
Besides intralayer valleys with nontrivial integer C v  , layer DOF can enrich QVHE. Considering a bilayer system with opposite ( ) C = ± ± lv 1 2 for upper and lower layers (the first sign ±) at K and K′ points (the second sign ±), thus, the -shape and V-shape valley states can simultaneously exist. Another strategy to increase DOF is F I G U R E 5 Hierarchal 1D arcs of DPs. (A) The first BZ and its projection on zigzag or bearded direction, containing two inequivalent DPs (K and K′). (B) Schematic of a 2D honeycomb lattice with three typical 1D boundaries, that is, armchair, zigzag, and bearded. (C) Projected 1D arcs related to two DPs at different boundaries. (D) Berry phase for zigzag or bearded boundary. 1D, one-dimensional; BZ, Brillouin zone; DP, Dirac point. artificial spins, which can be hybridized via fourfold degenerated double DP. [52][53][54] The fourfold degenerated node could be guaranteed by deliberate degeneracy of two 2D group representations or band folding mechanisms. Consequently, by manipulating spin-orbit coupling, acoustic QSHE with a pair of helical edge states is also possible in T-invariant systems. In this case, we can project them into spin space to obtain a pair of spin-Chern number C 1 = ± s . Next, we will focus on the acoustic realizations of the family of 2D QHE.
For acoustic QHE, its realization is not as straightforward as photonic QHE with the help of magneto-optical materials. The magnetoelastic effect usually is very weak and seems inaccessible, especially for airborne sound waves.
In 2014, the realization of the acoustic Zeeman effect associated with sound isolation in a circularly flowing background shed light on acoustic QHE. [27] Circular airflow acting as an effective magnetic field brings nonreciprocity breaking T symmetry for sound. The aeroacoustic model can be described as where ω represents angular frequency and ρ represents density. And A ⃗ = − periodical lattice, such as triangular [55] or honeycomb lattice, [56,57] acoustic models for QHE were theoretically proposed ( Figure 7A). Thus, the one-way chiral edge sound transport can be expected, robust against various defects. Flipping the airflow direction, the chirality of the edge state is inversed. Figure 7E plots the experimental setup of acoustic QHE and its chiral edge transport (anticlockwise in this case). [58] In a T-invariant system, the uniaxial gradient deformation of the graphene phononic crystal can induce a synthetic gauge field, which has been used to demonstrate acoustic Landau quantization and quantum-Hall-like edge states. [59] For acoustic QVHE, one typical case is to rotate triangular scatters ( Figure 7B). [60,63] C 3v reduces to F I G U R E 7 Acoustic models for the family of QHE. (A) Acoustic QHE based on circularly flowing background to break T symmetry. Reproduced with permission. [55] Copyright 2015, American Physical Society. (B) Acoustic QVHE by rotating triangular scatters to break P symmetry. Reproduced with permission. [60] Copyright 2017, Springer Nature. (C) Layer QVHE in a bilayer phononic crystal. Reproduced with permission. [61] Copyright 2018, American Physical Society. (D) Acoustic QSHE via manipulating the contrast ratio of honeycomb lattice to realize band inversion. Reproduced with permission. [62] Copyright 2016, Springer Nature. (E) Experimental setup of acoustic QHE, where inset shows measured acoustic pressure. Reproduced with permission. [58] Copyright 2019, American Physical Society. (F) Robust wave propagation for acoustic QVHE in a Z-shaped bend. Reproduced with permission. [60] Copyright 2017, Springer Nature. (G) X-shape splitter for acoustic QSHE. Reproduced with permission. [62] Copyright 2016, Springer Nature. QHE, quantum Hall effect; QSHE, quantum spin Hall effect.
However, in numerical calculations, the value of each C | | v is usually smaller than 1/2, coming from the overlap of two opposite valleys. The larger bulk bandgap the smaller the value. In the presence of T symmetry, the QVHE is less robust than the T-breaking QHE case. Nevertheless, valley states can still show good reflection-free performance against bends ( Figure 7F), [60] in which two opposite valleys will not scatter to each other. Directional acoustic antennas dependent on valley selection can also be realized when this valley state couples out into free space. [64] Furthermore, the width of the valley waveguide can be extended by inserting several layers of a Dirac phononic crystal to construct a sandwichlike configuration. [65] For the layer QVHE in a bilayer phononic crystal shown in Figure 7C, [61] the opposite rotating direction for upper and lower layers will give rise to C Δ 1 = ± lv , while the same rotating direction may cause C Δ 2 = lv acting as a double case of single QVHE. For acoustic QSHE, a typical acoustic model is a honeycomb lattice phononic crystal in which each primary unit cell contains two atoms ( Figure 7D). [62] The two 2D irreducible representations (E 1 and E 2 ) can be deliberately tuned to degeneracy at the center of BZ when the contrast ratio is chosen to be 0.3928 in this case. A larger or smaller contrast ratio will break such a fourfold Dirac node into two gapped twofold degenerated states. Amazingly, their bands are inversed with decreasing the contrast ratio (manipulating hopping), that is, lower-frequency two p states changed to upper frequency, indicating topological phase transition (TPT). An acoustic QSHE waveguide can be constructed at the boundary between ordinary and topological phononic crystals. And the helical edge states exhibit spinmomentum locking, robust against disorder, bends, and cavity. An X-shaped splitter can be introduced to check the artificial spins, where only cross-state is allowed while bar state is prohibited ( Figure 7G). Another conventional way to construct can resort to the band folding mechanism. [66] Considering a three times larger unit cell (containing six atoms) than the primary one in a honeycomb lattice, K and K′ DPs are folded in the center of BZ. Compressing or stretching inner atoms to change the intracell and intercell coupling can realize band inversion for acoustic QSHE as well. [67] Actually, the above-mentioned acoustic topological states are based on linear degenerated DPs. The quadratic, hybrid, or threefold degenerated may give rise to similar topological phenomena. [68,69]

| Artificial spins and fermionic pseudo-TRS
In electronic systems, the spin-momentum locking of QSHE is protected by fermionic T symmetry (T 1 = − f 2 ), naturally guaranteeing the Kramers degeneracy. The corresponding symmetries in acoustic QSHE need to be examined because of the different intrinsic spins between fermionic electrons (spin-±1/2) and bosonic phonons (spin-0). [70,71] As shown in Figure 8A, a pair of acoustic spins are hybridized via symmetry (S) and antisymmetry (A) states as S ± iA at the boundary. [62] They can further refer to the bulk Bloch states d p + x y x + 2 2 and d p + xy y due to their different parity ( Figure 8B). Under C 6v symmetry, for acoustic waves, we can redefine a fermionic pseudo-TRS where τ v and τ d denote two orthogonal mirror operators, σ y denotes the y-component of the Pauli matrix, and K denotes the complex conjugation (T operator of spin-0 phonon). The S state under T f operation changes to A, one more to −S state. It means that two acoustic spins can act as artificial spin-±1/2 particles associated with fermionic pseudo-T symmetry with . In other words, S and A states can be treated as electric (E) and magnetic (H) fields accompanying electromagnetic ). The corresponding fermionic pseudo-T can also be obtained as T σ σ K iσ K = =− f z x y . However, such C 6v -based T f is hardly kept at the 1D boundary. A mini-gap for a pair of acoustic helical edge states will appear. Therefore, acoustic backscattering is heavily suppressed but not completely immune. The reflection-free defect types and strengths are limited as well. The mini-gap can be decreased even totally eliminated by enforcing additional boundary symmetries or modifying boundary shapes.
In fact, the lattice (crystalline) symmetry-enabled acoustic analog of QSHE is classified into a fragile topology. [72] Bands with fragile topology mean that additional trivial Bloch bands would trivialize those topological ones, though the topological set is separate from the trivial one in usual cases. In contrast, the stable phase can only be trivialized by another topological band. For example, if we impose s states into p/d constructed helical edge states, their T f character and nearly gapless behavior will disappear. Such fragile topology can be checked by the Wilsonloop method [73] or elementary band representations method under twisted boundary conditions. [74] Furthermore, breaking artificial spin conservation via synthetic spin-orbit coupling can realize the nonpaired acoustic spin-Chern insulator. [75] Recently discovered acoustic Möbius insulators empowered by projective PT-symmetry with PT 1 ( ) = − 2 in the presence of gauge DOF. [76][77][78] Figure 9A illustrates a model with negative and positive hopping amplitudes, keeping both primitive translational symmetries L x,y . And each plaquette hosts a π gauge flux. The band structure has a fourfold DP at the corner of BZ. Only breaking L y , it can be gapped in which a pair of edge states appear degenerated at k π = ± x point ( Figure 9B). Such edge states will experience 4π periodicity in momentum space, deemed as Möbius twist. Figure 9C is a schematic of a 2D experimental acoustic lattice. The positive and negative couplings are designed via various connecting configurations between two acoustic dipole modes. [79] This strategy can also be extended to 3D geometry to realize higher-order 1D hinge states.

| Three-dimensional acoustic TIs
In this section, we focus on the generalization of 2D acoustic topological states to 3D. Up to 230 types of space groups could advance various kinds of acoustic topological states due to their flexible and designable units and structures. Intuitively, directly stacking 2D QSHE layers along the z-direction with negligible interlayer couplings can get two crossing 2D surface state planes ( Figure 10A). [80] Their spinmomentum locking still exhibits in the xy plane, while at each k z slice, they are two flat in-gap surface bands. Another construction method is based on 1D SSH F I G U R E 9 Acoustic Möbius insulators based on projective symmetry. (A) Both primitive translational symmetries L x,y are preserved, hosting a fourfold degenerate Dirac point. The red (blue) bonds denote negative (positive) hopping. (B) The staggered dimerization only preserves L x , hosting Möbius twist for edge states. (C) Schematic of lattice structure for an acoustic Möbius insulator. Reproduced with permission. [78] Copyright 2022, American Physical Society. chains to shape a NaCl-like 3D structure ( Figure 10B). [81] The original zero-dimensional (0D) edge states will expand into 2D in momentum space. Considering there exists a glide symmetry for a 2D domain-wall, for example, G y : (x, y, z) → (x+ a/2, −y, z), we have G y 2 : (x, y, z) → (x+ a, y, z  Figure 10C). [82] The neighboring layers have opposite valley Chern numbers for a pair of acoustic artificial spin ±, at H point C ( ) ∓ /2. Thus, the acoustic pseudospinvalley Chern numbers can be described as based on band inversion is shown in Figure 10D. [83] One layer contains uniform atoms, compressing or stretching it like a 2D QSHE case. The other layer behaves like a QVHE case. The structure belongs to noncentrosymmetric space group No. 189 (P 6 2m). The surface states are shaped in a semi-Dirac cone, that is, linear dispersion in the layer plane while quadratic out of the layer plane.
For photons and phonons, the z-direction P-breaking layered tetragonal or triangular lattices have been proposed to support quadratic or linear surface Dirac cones on their top facet or lateral Jackiw-Rebbi domain wall ( Figure 10E). [84,85] However, they are difficult to realize in phononic crystals due to the small bandgap or impractical hopping parameters. The chiral interlayer coupling could mediate these problems to open a relatively large full bulk bandgap ( Figure 10F). [86] Notably, strong TI is impossible in bosonic systems due to lacking intrinsic spin-±1/2. Acoustic TI belongs to either weak TI or crystalline one. Therefore, the 2D surface Dirac cones appear in pairs. The quadratic cone can be treated as two linear ones meeting at a high-symmetry point in momentum space, for example, the BZ center.
Here, we would like to give an example of acoustic TIs. [86] Like 2D QSHE case, one can start begin with 3D Dirac cones that are linear dispersions of the bulk band structures along all three directions. As illustrated in ( Figure 11A), by introducing effective spin-orbit coupling, a full topological bandgap could be created, supporting 2D surface Dirac cones with massless Dirac quasiparticles at the surfaces. Further lifting these 2D surface Dirac cones by breaking P symmetry may create a 1D topological edge of the 2D surface, which is a 3D secondorder TI with 1D gapless hinge states. Figure 11B shows a BZ folding band breaking along the z-axis. Accordingly, the space group of phononic crystal changes from No. 173 (P6 3 ) to No. 168 (P6). After this process, both acoustic spin DOF and 3D full bandgap are obtained. Figure 11C plots topological surface states on both lateral and top facets. For the lateral facts, we can create a 2D topological domain wall in the yz plane (reversed z-direction with opposite sign of effective mass). As a result, the surface Dirac cone can be observed, acting as acoustic analogy of quantum relativistic Jackiw-Rebbi states. For the top fact, only a hard boundary condition (complete reflection covered by a plastic board) is enough to support acoustic topological crystalline states. The measured 2D There are two twofold degenerated surface Dirac cones near the projection of K and K′ in the whole top surface BZ. Further utilizing such two 2D surface Dirac cones can construct a 3D second-order acoustic TI. Folding BZ in the xy plane, we can tune these two cones to form a double-surface Dirac cone at the BZ center by considering a three times larger unit cell. Compressing or stretching honeycomb lattice leads to a full surface bandgap associated with a band inversion. In this case, the original bulk DP is eightfold degenerated node. Further breaking these hinge states with a bandgap may hierarchically bring a third-order corner state. [ 87] In real 3D solid-state materials, dislocations are unavoidable defects, such as edge and screw dislocations. Their Burgers vectors are always integer values, probably possessing some real-space topological character. Figure 12A demonstrates a screw dislocation running spirally in a layer-stacked acoustic topological model. [88] A network of coupled acoustic ring resonators acts as a single layer for stacking ( Figure 12B). The inset shows the tilted interlayer coupler by considering the periodicity of the system along the z-axis. The dislocation-projected band structure is shown in Figure 12C, where a pair of 1D helical topological dislocation states can be obtained in a 3D geometry. In contrast to the conventional BBC, the so-called bulkdislocation correspondence is directly verified in a 3D acoustic model. Like glide symmetry, screw symmetry enforces the democracy of a pair of helical states at the BZ boundary k π = ± z . For example, a step screw dislocation at the center of a 3D phononic crystal ( Figure 12D,E) is related by the screw rotation S x y z : ( , , ) [89] After dimensional reduction, the 3D system can be mapped into various k z -dependent 2D slices. Each coupling between neighboring quarter sectors has a gauge phase ±k z h/4. Thus, an artificial gauge flux is successfully inserted into an acoustic model. According to Bloch's theorem, this artificial gauge flux varies from 0 to 2π in a full circle. Figure 12F shows the topological Wannier cycles induced helical screw dislocation states.

| Acoustic TSMs
TSMs are gapless phases hosting nontrivial band degeneracies, [90] including 0D nodal point degeneracies, 1D nodal lines, and 2D nodal surfaces. The touching bands may possess different quantum numbers, which avoids the band repulsion. The Weyl point is a doubly degenerate point and carries a nontrivial topological charge ( Figure 13A). [41] The surface states form a Fermi arc, linking a pair of Weyl points in momentum space. The 3D DP is a fourfold degenerate point, and can be viewed as a combination of two Weyl points with opposite charges ( Figure 13B). [91] The nodal line is a 1D band crossing and exists in the presence of certain crystalline symmetries ( Figure 13C). [92] It has various types, including nodal rings, nodal links, and nodal chains. The 2D band crossing is called a nodal surface. The nodal line and nodal surface can also carry a topological charge.
Weyl fermions, originally predicted by Hermann Weyl to describe massless fermions, have recently been discovered in Weyl semimetals with linear dispersions in the 3D momentum space touching at a degenerate point-the Weyl point. The low-energy excitations around the Weyl point can be described by the Weyl Hamiltonian where v i are the group velocities, and σ i are Pauli matrices. The Weyl Hamiltonian contains all three Pauli matrices, as a result, Weyl points are robust against perturbations and can only be annihilated in pairs with opposite chirality. They can be viewed as monopoles of Berry flux in momentum space and carry a nonzero topological charge (Chern number), and the sign of the charge corresponds to the chirality of Weyl points. Since the Berry curvature will vanish when P symmetry and T symmetry are both kept, one of them needs to be broken to realize Weyl points.
In acoustics, Weyl points were experimentally observed in 3D chiral phononic crystals with broken P symmetry. [93,94] The phononic crystals are constructed by stacking 2D honeycomb lattice layers and introducing a chiral interlayer coupling using slanted waveguides ( Figure 14A). Two pairs of Weyl points with charge ±1 are located at the highsymmetry points ( Figure 14B) and linear dispersions are exhibited. In the momentum space, a 2D plane with a fixed k z can acquire a net Berry flux due to the existence of Weyl points, resulting in a nonzero Chern number for the 2D band, which indicates the existence of topologically protected one-way states at the surface. The equifrequency curves of the surface states near the Weyl point frequency form open arcs, called the Fermi arcs, that connect the projections of two Weyl points with opposite chirality ( Figure 14C). The unique Fermi arcs can result in interesting phenomena, such as topological negative refraction [95] ( Figure 14F) and sound collimation. [96] The type-II Weyl points have also been proposed and realized in acoustics, by stacking dimerized chains of acoustic resonators [99] or graphene-based layers. [100] The type-II Weyl points violate the Lorentz symmetry and cannot exist in high-energy physics. The dispersions around the type-II Weyl points are strongly tilted, leading to an open hyperboloid iso-energy surface near the Weyl point. In nonsymmorphic crystal structures, there also exist unconventional band degeneracy points beyond the conventional Weyl point and DP, such as spin-1 Weyl points [97,101,102] and quadratic Weyl points, [98] which are protected by lattice symmetries and carry higher topological charges. The spin-1 Weyl points are formed by three bands ( Figure 14D), two linear bands with an additional flat band, which carry topological charges of (−2, 0, 2). The charge-2 quadratic Weyl points are twofold degenerate and exhibit quadratic dispersions in two directions and a linear dispersion in the third direction ( Figure 14E). A hallmark of the charge-2 Weyl points is the existence of double Fermi arcs. Synthetic dimensions can be utilized to realize Weyl points in low-dimensional physical systems. By using two extra structural parameters, synthetic acoustic Weyl points have been observed in a simple 1D phononic crystal. [103] The synthetic Weyl points can also be related to the 2D valley physics. [104] F I G U R E 13 Schematic of band degeneracies in acoustic TSMs. The 3D DP is a fourfold linearly degenerate point, and can be realized by band inversion [105] or the symmetry-enforced mechanism. [106] The 3D DP is equivalent to a combination of two Weyl points with opposite charges, and thus it carries a net-zero topological charge. It can be treated as a topological transition point between different phases. By breaking certain symmetries, the splitting of 3D DP may give rise to TIs or Weyl and nodal line semimetals. The transition of DPs toward Weyl points has been reported in acoustics. [105] The presence of surface states is subtle in the Dirac semimetal due to the zero topological charge and the bulk-surface correspondence may be absent. Recently, the bulk-hinge correspondence was demonstrated in the acoustic Dirac semimetal, revealing the fundamental relation between higher-order hinge arcs and 3D DPs. [107,108] In the 3D band structure, two bands can cross each other at closed lines, called the nodal lines. The ring-like nodal lines were observed in a layer-stacked phononic crystal. [109] The existence of the nodal line is protected by the mirror symmetry. The Berry phase along a closed loop that encloses the nodal line is π, which leads to the drumhead surface states with exotic transport properties. Two bands can also touch a 2D surface, called the nodal surface. The dispersion is linear in the vicinity of the surface. The nodal surface stabilized by a nonsymmorphic symmetry was demonstrated in a 3D acoustic crystal, [110,111] which carries a topological charge of 2. Two Weyl points appear together with the nodal surface and a pair of open Fermi arcs connect them in the momentum space.

| HIGHER-ORDER TOPOLOGICAL INSULATORS
Over the past few years, a new type of topological phases of matter has emerged and attracted tremendous attention, known as the higher-order topological insulators (HO-TIs). [112,113] Compared with conventional TIs, HOTIs feature gapless topological boundary states in lower dimensions. For example, 2D HOTIs host gapped 1D edge states and in-gap 0D corner states. In three dimensions,  [93] Copyright 2015, Springer Nature. (C) The Fermi arcs connect the projections of two Weyl points with opposite charges. Reproduced with permission. [94] Copyright 2018, Springer Nature. (D) The spin-1 Weyl points are formed by three bands. Two linear bands and an additional flat band intersect at the Weyl point. Reproduced with permission. [97] Copyright 2019, Springer Nature. (E) The quadratic Weyl points with quadratic dispersions in two directions and a linear dispersion in the third direction. Reproduced under terms of the CC-BY license. [98] Copyright 2020, The Authors, published by Springer Nature. (F) Topological negative refraction on the surface of the Weyl semimetal. Reproduced with permission. [95] Copyright 2018, Springer Nature.
HOTIs can either manifest a hierarchy of 2D surface states, 1D hinge states, and 0D corner states, or only support gapped surface states and gapless hinge states, depending on different intrinsic and extrinsic symmetries (see Figure 15 for a schematic illustration of HOTIs in different dimensions). With the multidimensional topological states, HOTIs have not only generalized the conventional BBC and offered broader and deeper understanding on the topological band theory, but also exhibited great potentials in photonics and phononics, such as robust multidimensional transports/localizations in higher-dimensions, topological cavity modes with high quality factors, multidimensional topological control, and energy harvesting.
In this section, we review this fast-growing field by starting with an introduction of the fundamental principles and theories of the HOTIs, followed by the experimental demonstrations. Then we turn to the latest advances that have moved the studies of HOTIs further toward even more interesting topological phenomena, including the gapless higher-order topological semimetals (HOTSMs) and HOTIs enriched by extra DOFs, such as synthetic dimensions, non-Hermiticity, nonlinearity, and aperiodicity or quasicrystalline symmetries.
It is pointed out that though our main perspectives are focused on the acoustic and phononic systems, other developments in photonics or electric circuits will be mentioned as well to inspire the acoustic and phononic communities.

| Fundamental principles and theories of HOTIs
Historically, attempts to explore lower-dimensional topological boundary states from higher dimensions have been made way before the emergence of the terminology HOTIs. It was shown that subject to magnetization, 2D topological surface states of 3D ferromagnetically ordered materials became gapped and 1D gapless chiral edge channels appeared with the Hall conductance topologically quantized. [114,115] These were the early witnesses for the higher-order topological phenomena. More recently, even lower-dimensional topological boundary states, for example, corner states, were discovered in topological crystalline insulators, which fundamentally opened a new chapter for HOTIs. Starting from the pioneering work by Benalcazar et al., [113] the dipole polarization has been generalized to higher electric multiple moments, such as quadrupole, octupole, and even hexadecapole (16-pole) that support a hierarchy of multidimensional topological boundary states. Later on, HOTIs based on nontrivial dipole polarizations [116][117][118] and multidimensional TPTs [119] were identified. In these developments, versatile crystalline symmetries have exhibited great potential facilitating the demonstrations and explorations of the HOTIs. Particularly assisted by the photonic and phononic materials with artificial designs, well-controlled material processing and mature characterization techniques, this field has led to explosive growth during the past few years. The basic principles and mechanisms for HOTIs in crystalline insulators, that is, the multipole moments, the nontrivial dipole moments, and the multidimensional TPTs as mentioned above, will be reviewed in the following. Higher-order topological phases of matter based on other principles will be discussed in Section 4.3.

| Topological multipole insulators
The multipole moments are physical generalizations of the dipole polarizations. [120] According to the modern theory of polarization, [121,122] a bulk dipole moment manifests itself through the existence of boundary charges (see Figure 16A). In conventional TIs, these essentially are the topological edge states. When considering one dimension higher, a macroscopic material allows charge density in multiple dimensions, which constitutes either 1D/2D dipole moments or a quadrupole moment, as schematically illustrated in Figure 16B. An intuitive extrapolation of the manifestations of the quadrupole moment in a finite material with boundaries is F I G U R E 15 Schematic illustrations of HOTIs in two and three dimensions. 3D, three-dimensional; HOTI, higher-order topological insulator.
the emergence of corner localizations. This turns out to be indeed the case for a quadrupole insulator hosting corner states. [113] It should be pointed out that usually the multiple moments are defined with respect to a particular reference frame. Therefore, in order for the quadrupole moment to be independent of the choice of reference frames, the dipole moments must vanish (a direct generalization is that in order for any higher moments to be well defined, all the lower moments must vanish). [120] In this sense, a quadrupole insulator always accompanies gapped edge states and the corner states fall into the edge gap. Following a similar analysis, the multipole theory can be generalized to even higher dimensions and higher moments, such as the octupole in 3D systems and hexadecapole in four-dimensional (4D) systems.
To realize a quadrupole insulator, the minimal tightbinding model has been proposed, consisting of a square lattice of four sites in each unit cell with both positive and negative nearest-neighbor couplings ( Figure 16C). Such a configuration guarantees two noncommutative reflection symmetries, which are responsible for canceling the dipole moments and giving rise to the nontrivial quadrupole moments. [113] A recent study showed that in addition to manipulating the couplings, the nonsymmorphic space symmetries can also be exploited to realize nontrivial quadrupole moments by probing the topological properties in the second bandgap. [123] This idea was implemented in an acoustic artificial crystal (see Figure 16D for the lattice design), which enabled an interesting hierarchy of topological multipoles, that is, the first bandgap hosted nontrivial dipole moments while the second bandgap hosted nontrivial quadrupole moments (with canceled dipole moments). Regarding the characterizations of the nontrivial quadrupole (and higher) moments, the nested Wilson-loop approach [113,120] or complementarily the many-body order parameters [124,125] are often used in practices. The tight-binding model (Reproduced with permission, [113] Copyright 2017, The American Association for the Advancement of Science) and (D) the artificial crystal with nonsymmorphic space symmetries to realize nonzero quadrupole moments (Reproduced under terms of the CC-BY license, [123] Copyright 2020, The Authors, published by Springer Nature). (E) Filling anomaly identified by Wannier representations with C 4 -symmetry (Reproduced with permission, [118] Copyright 2019, American Physical Society), which can be implemented in (F) a 2D SSH lattice (Reproduced with permission, [117] Copyright 2018, American Physical Society). (G) Diagram of the multidimensional TPTs. 1D, one-dimensional; HOTI, higher-order topological insulator; SSH, Su-Schrieffer-Heeger; TPT, topological phase transition.

| HOTIs with dipole moments
Conventionally, crystalline insulators characterized by dipole moments were considered to host gapless edge states and hence no corner states. However, it was later identified that in an insulator that allows a Wannier representation (which is intimately related to the quantization of the dipole moments), [126] a mismatch between the Wannier centers of the occupied bands and the atomic positions in the crystal occurs. [127] In other words, there exists electron charge filling anomaly in such an insulator. Enforced by intrinsic crystalline symmetries, the filling anomaly often appears in geometric corners, consequently leading to nonzero corner charge that is manifested as the corner state. This essentially indicates that the insulator is indeed a HOTI. Figure 16E depicts the fractional charge distributions in a crystalline insulator with C 4 symmetry, whose Wannier centers are indicated by the red dots. The quarter portion of the Wannier orbital in each corner cell accounts for the corner charge. A more systematical analysis on the corner charge anomaly and the associated higher-order topological properties in various crystalline insulators with C n -symmetries have been provided by Benalcazar et al., [118] offering a fundamental theory for the HOTIs with dipole moments. It should be pointed out that Benalcazar et al. [118] argued that in order for the corner anomaly to be well defined, both the bulk and edge are required to be insulating, suggesting zero bulk polarization and hence trivial dipole moment (which can be realized by combining two insulators with nontrivial dipole moments). Subsequent studies, on the other hand, showed that the corner states can be stabilized even if there is only bulk bandgap [117] or even in cases where the corner states are coupled to or hybridized with the bulk or edge states. [128] These instances happen in artificial classical systems where the notion of charge density is replaced by the mode density which is free of constraints imposed by the electronic charge neutrality.
Due to the high-dependence on crystalline symmetries, the HOTIs with fractional corner charges can be realized in various C n -symmetric lattices, where the topological properties are controlled by modulating the intra-and intercell couplings. For example, a 2D extension of the SSH model (see Figure 16F) with intercell couplings stronger than intracell couplings enables nontrivial dipole moments along both the xand y-directions, [117] giving rise to the Wannier configuration exactly matching the one shown in Figure 16E. Consequently, four corner states that are localized at the four geometric corners emerge, as the defining signatures of a HOTI. The characterizations of this type of HOTI mainly rely on the Wannier representations. More details can be found in Benalcazar et al. [118] and Marzari et al. [126]

| HOTIs based on multidimensional TPTs
The above-discussed principles for HOTIs are based on the bulk topological properties. A recent recognition showed that by independently exploiting the bulk and edge topological properties, an interesting multidimensional hierarchy of topological phenomena can be realized, that is, the bulk TPT gives rise to the edge states while the edge TPT leads to the emergence of corner states. [119] In this sense, the higher-order topology is solely determined by the edge properties and is independent of the bulk properties (as long as the bulk is insulating). The diagram of these multidimensional TPTs and the occurrences of the edge and corner states are schematically illustrated in Figure 16G.
This scheme has been implemented in a 2D acoustic crystal respecting the glide symmetry, which enabled independent controls of bulk and edge topological properties by tuning the lattice geometries. [119] Specifically, by gapping a DP in the 2D bulk band structure, the bulk TPT occurred and gapless edge states emerged at the interfaces between crystals in different topological phases. By further tuning the edge/interface geometries, the gapless edge states were gapped. Within the edge gap, topological corner states appeared as consequences of the nontrivial edge topology. More recently, similar topological phenomena were observed in 3D systems, where the nontrivial bulk topology granted the gapless surface states while the nontrivial surface topology accounted for the gapless hinge states. [86] Due to the independent topological properties in the bulk and edge/surface, this type of HOTI can be characterized following the typical methods for traditional TIs. For example, in Zhang et al., [119] the band inversion and associated Dirac masses with reversed signs were used to characterize the nontrivial bulk and edge topological properties.

| Experimental demonstrations
As mentioned above, the classical systems have exhibited great potentials in demonstrating and exploring various HOTIs due to their highly feasible artificial designs, manufacturing, and measurements. In fact, most of the experimental implementations and observations of the HOTIs have been reported in classical systems, while only a few naturally occurring HOTI materials were identified. [129,130] In this section, we give a quick exhibition of the experimental progress of the HOTIs, with a particular emphasis on the acoustic and phononic implementations.
Following the theoretical developments, the experimental progress exhibits a similar pattern. The early experiments were focused on realizing the quadrupole insulators. A seminal work was done by Serra-Garcia et al., [131] who implemented the model in Figure 16C using a phononic crystal consisting of silicon plates connected by thin bent beams. The out-of-plane vibration was considered and the positive and negative couplings were mediated by elaborately designing the beam connections. Figure 17A1 illustrates the phononic crystal design. The experimental setups and the measurements on the bulk and edge/corner states are shown in Figure 17A2,A3. Around the same time, an implementation in a photonic microwave system was also reported. [132] Moving forward, clever designs of positive and negative couplings were proposed and realized in airborne systems, [79,133] as displayed in Figure 17A4,A5. Artificial acoustic crystals respecting nonsymmorphic space symmetries, such as p4g wallpaper group, were also designed and constructed to realize the nontrivial quadrupole moment, [123] offering alternative perspectives Reproduced with permission. [131] Copyright 2018, Springer Nature. (A4, A5) Designs of positive and negative couplings in airborne systems. Reproduced with permission. [79,133] Copyright 2020, Springer Nature and Copyright 2020, American Physical Society. (A6-A8) Realizations of higher octupole and hexadecapole insulators. Reproduced with permission. [79,134,135] Copyright 2020, Springer Nature, and Copyright 2020, American Physical Society. (B) Various experimental implementations of dipole-moment-enabled HOTIs. Reproduced with permission. [136][137][138][139][140][141] Copyright 2019, 2021, American Physical Society, and Copyright 2019, 2020, Springer Nature. (C1-C3) Respectively, the illustration of the occurrence of corner states based on edge TPTs, a unit cell of the designed acoustic crystal, and the corresponding experimental measurements. Reproduced with permission. [119] Copyright 2019, Springer Nature. (C4, C5) Surface TPTs in a 3D acoustic crystal and the occurrence of gapless hinge states. Reproduced under terms of the CC-BY license. [86] Copyright 2020, The Authors, published by Springer Nature. HOTI, higher-order topological insulator. and material platforms for multipole insulators. Meanwhile, higher moments were also demonstrated, including the octupole insulators in 3D acoustic crystals [79] and the hexadecapole insulators in synthetic spaces [134] or in 4D electric circuits. [135] The corresponding designs and experimental setups are shown in Figure 17A6-A8.
For the HOTIs with dipole moments, various C n -symmetries and the experimental observations of their resultant higher-order topology have been greatly facilitated by the artificial designs, especially owing to the versatile manipulations on couplings via either cavity-tube configurations or scattering-type crystals (see Figure 17B). These demonstrations have been given in SSH-like lattices, [136] breathing Kagome lattices, [137,138] triangular lattices, [139] honeycomb lattices, [140] and so forth. Specifically for 3D cases, the HOTIs exhibit an interesting dimensional hierarchy of the topological boundary states. That is by decreasing the dimensions, one can successively observe the 2D surface states, 1D hinge states, and 0D corner states, manifesting a special multidimensional bulk-surfacehinge-corner correspondence. [141][142][143] As discussed in Section 4.1.3, the HOTIs based on multidimensional TPTs have been implemented in artificial acoustic crystals with glide symmetry in 2D systems [119] and in a bilayer honeycomb design in 3D systems. [86] The interesting behavior is that it is solely the edge/surface topological properties that determine the higher-order corner/hinge states. As shown in Figure 17C1, in the 2D HOTI (whose unit cell is illustrated in Figure 17C2), the Dirac masses on the x-and y-edges have opposite signs generating a domain wall at the boundary of the boundaries. This corresponds to the emergence of the corner states ( Figure 17C3). For the 3D HOTI, the special bilayer design enables the control of surface TPT without affecting the bulk gap (see Figure 17C4). As a result, the gapless hinge states emerge in the surface gap (see Figure 17C5), signifying the higherorder topological features in such a 3D system.

| Other higher-order topological matters
In previous sections, we have focused on the main frame of HOTIs based on crystalline symmetries. There have been other developments of higher-order topological matters. Especially with the well-established HOTI theories and relatively mature experimental practices, the latest advances have brought the studies of higherorder topology into a next level where the HOTIs are enriched by extra DOFs, such as synthetic dimensions, non-Hermiticity, nonlinearity, and aperiodicity or quasicrystalline symmetries. In addition to the gapped phases, the gapless higher-order topological matters, that is, the HOTSMs, have attracted tremendous attention. In this section, we review these developments.

| HOTIs in synthetic dimensions
The development of synthetic dimensions has offered novel routes to facilitate the designs of topological phases of matter. The main idea is to couple certain nonspatial states spanned by an internal DOF, for example, the frequency, polarization, orbital angular momentum, temporal modulation, or even geometric/material parameters, to form discrete dimensions. The creation of one or more such artificially synthetic dimensions allows for the study of higher-dimensional physics in lowerdimensional structures. Sometimes, it is even possible to go beyond the 3D Euclidian space. Accompanying the developments of HOTIs, the concept of synthetic dimensions has been quickly brought into this field.
Owing to their flexible and versatile modulations and coupling controls, the photonic and electric circuit communities have led the way by showing how higher multipole moments such as a hexadecapole moment can be realized in synthetic frequency dimensions [134] or in 4D electric circuit lattices formed by connecting circuit nodes and arranging unit cells along eight directions, [135] which otherwise were elusive in real spatial dimensions. The acoustic community has also skillfully proposed clever ways to construct synthetic dimensions. Chen et al. [144] showed that a 4D hexadecapole insulator can be mapped to an aperiodic 1D acoustic cavity array (the aperiodicity is controlled by modulating the couplings between the adjacent cavities) via the Lanczos transformation, suggesting the latter has encoded the information of the 4D insulator into the aperiodicity as the synthesized dimensions. Figure 18A1 depicts the tight-binding model for the 4D hexadecapole insulator (upper panel) and its mapping to a 1D acoustic crystal of cavities with sophisticated couplings (lower panel). Chen et al. [145] realized synthetic dimensions by modulating the resonant frequencies of the on-site acoustic cavities (see Figure 18A2). Together with the two spatial dimensions, they created a 4D lattice which was shown to support both (4 − 1)D hypersurface states and (4 − 2)D higher-order surface states. In addition to the static studies, it has been shown that by incorporating synthetic controls and modulations, higher-order TP, [146] and Floquet corner states that carry a similar oscillating behavior as the time crystal modes can be realized. [147] 4.3.2 | HOTIs enriched by non-Hermiticity, nonlinearity and aperiodicity/quasicrystalline symmetries Another research trend in the field of HOTIs is to explore the interplay between higher-order topology and extra physical DOFs, such as non-Hermiticity, nonlinearity, aperiodicity/quasicrystalline symmetries, and so forth. In Gao et al., [148] it has been shown that a nontrivial higherorder quadrupole moment can be realized by deliberately introducing loss into an artificial acoustic crystal. Solely depending on the loss configurations, the acoustic bandgap can be either topological or trivial. This is different from the Hermitian systems where the crystalline symmetries are exclusively responsible for the nontrivial higher-order topology. On the basis of this lossy design, typical topological features of a quadrupole insulator, that is, gapped edge states and in-gap corner states have been experimentally observed, as illustrated in Figure 18B1. Using loss as an ingredient to control the higher-order topology was also explored in electric circuits, [154] where the resistors were used to introduce loss and accordingly open bandgaps hosting nontrivial higher-order topology. When the non-Hermitian manipulation carries a non-Bloch feature, for example, asymmetric couplings or nonreciprocity, it can interplay with the higher-order topology in unprecedented ways. For example, a recent study showed that by skillfully designing the loss configurations in the acoustic crystal composed of ring resonators ( Figure 18B2, left panel), asymmetric couplings can be introduced to give rise to an interesting non-Hermitian phenomenon dubbed higher-order non-Hermitian skin effect (NHSE). [149] This effect has been shown to fundamentally alter the higher-order topology by dictating all the acoustic modes allowed in the system to accumulate toward specific geometric corners. This is in stark contrast to the Hermitian HOTIs ( Figure 18B2, right panels).
In addition to non-Hermiticity, nonlinearity has also been found to enrich the higher-order topology. For example, it was shown that the nonlinear effect of a homogeneous global pump can switch a trivial insulator into the HOTI phase, [150] as schematically illustrated in Figure 18B3. By introducing nonlinearity, nonlinear topological corner states and the formation of corner solitons were observed. [155] A square-root HOTI was realized based on a nonlinear photonic crystal. [156] By incorporating nonlinearity, higher-order bound states in continuum were observed, which exhibited nonlinear coupling with topological edge states. [157] So far, the nonlinearity-enriched HOTIs have been limited to photonic systems, while the possibilities in acoustic and phononic systems are still open to explore, likely facing the technical challenges to incorporate the nonlinearity.
Recently, it has been recognized that not only the periodic lattices can give rise to nontrivial higher-order topology, but also the aperiodic/quasicrystalline systems are fertile playgrounds for HOTIs. Disorders have been well known for their intriguing effects on conventional TIs, as identified in topological Anderson insulators. It turned out that strong disorders also have nontrivial effects on the higher-order topology and can lead to the formation of topological corner states. [151] This process has been demonstrated in an electric circuit design, as shown in Figure 18B4. In addition to the disorders, other types of aperiodicity such as aperiodic Kekule modulations have exhibited potentials to generate corner states in arbitrary geometries, [158] and Ammann-Beenker tiling quasicrystals have been shown to support a nontrivial quadrupole moment. [159]

| HOTSMs
With the great success of the gapped HOTIs, it is natural to extend the studies into the gapless systems where the HOTSMs have emerged as novel higher-order topological phases. These materials are featured with Weyl points and DPs in the 3D bulk band structures, which are connected by 1D hinge arcs, in comparison to the conventional TSMs where the Weyl points and DPs are often connected by 2D surface arcs. Recently, the Weyl HOTSMs have been realized via 3D chiral stacking of 2D HOTIs [152] or by designing tetragonal lattices with uniaxial screw symmetry in artificial acoustic crystals. [153] Therein, hinge states were experimentally observed to connect the projections of the Weyl points with opposite topological charges along the stacking direction ( Figure 18C1,C2).
In addition to the Weyl HOTSMs, the Dirac HOTSMs have also been explored. A 3D DP represents a degenerate pair of Weyl points with opposite chirality and is rare in naturally occurring materials. However, facilitated by versatile artificial designs, classical photonic and acoustic systems have shown high ability to construct DPs in three dimensions. Further enabled by certain crystalline symmetries, these materials have exhibited gapless higher-order topological features, that is, the 3D bulk band structures accommodate DPs and their momentum-space projections are connected by 1D hinge states (see Figure 18C3 for an example). [107,108,160] Especially, Xia et al. [107] argued that in the Dirac HOTSMs, the BBC should be deemed bulk-hinge correspondence, where the 1D hinge states directly correspond to the nontrivial 3D bulk topology while the 2D surface states are removable, suggesting the failure of the bulk-surface correspondence.

| NON-HERMITIAN TOPOLOGICAL PHYSICS
The Hermiticity of a Hamiltonian guarantees the selfadjointness of a valid observable in quantum mechanics for a conservative system. However, real physical systems always involve energy exchange and probability nonconservativeness due to the interactions with the environments, leading to the breakdown of Hermiticity. Contrary to the traditional impression that the non-Hermitian effects such as energy loss are always harmful, recent studies have shown that the considerations of non-Hermiticity can bring forward novel physics, such as non-Hermitian parity-time (PT) symmetry, exceptional features and novel non-Hermitian topological matters. Owing to the feasible manipulations of gain/loss and active controls, artificial photonic, acoustic, and phononic systems in particular have inspired versatile non-Hermitian phenomena, leading to great success in both fundamental physics and potential applications.
In this section, we review the recent progresses in this field with an emphasis on the non-Hermitian topological phenomena that cover two essential subjects. The first subject is the exceptional degeneracies and their topological properties. We begin with a brief survey of the PT-symmetry and exceptional points (EPs) in the parameter spaces, and then focus on the exceptional features enabled by periodic lattices where point, ring, nodal, knotted, and linked non-Hermitian structures have emerged in the momentum space as novel gapless topological phases. The associated Fermi arcs are also talked about. For the second subject, we start by briefly summarizing the definitions of the non-Hermitian bandgaps and their classifications. Then we discuss the consequences of non-Hermiticity on boundary states of gapped topological phases of matter. Therein, the effect of on-site gain/loss is first reviewed. This is in an exhibition of the novel topological features enabled by the on-site gain/loss. In particular, the topological lasers and sasers (the acoustic version of lasers) are discussed. Then, we move to the novel notions of non-Hermitian topological phases that are built on the induction of asymmetric couplings, where the topological properties of single-band systems, NHSE, breakdown of conventional BBC, and non-Bloch PT-symmetry and exceptional features as particularly fascinating and unique non-Hermitian phenomena are reviewed.
Note that to be clearly organized, the contents of the two aforementioned subjects are sometimes separated into different subsections, which, however, follow the above logic flow.

| PT-symmetry and EPs in the parameter spaces
A great proportion of studies on non-Hermitian physics have been constituted by the non-Hermitian PT-symmetry and EPs. It has been pointed out that real eigenvalues are not necessarily the results of Hermiticity, but can also be realized if the system is non-Hermitian but obeys PT-symmetry. [161][162][163] PT-symmetry challenges the standard convention that physical systems have to be Hermitian and allows the studies of new kinds of non-Hermitian physics with no Hermitian counterparts. One important feature of PT-symmetric systems is the transition from real to complex eigenenergies when the non-Hermiticity is modulated, known as the PT phase transition. [162] Associated with this transition, the physical system goes from equilibrium (often referred to as the unbroken PT-symmetric phase) to nonequilibrium (i.e., the PT-broken phase). [164] The transition point corresponds to the so-called EP. At the EP, both the eigenvalues and eigenvectors of the system's Hamiltonian coalesce, forming singularities. [165] Such a mathematical curiosity has soon become a physical reality in artificial classical systems where cleverly balanced gain and loss have been used to realize PT-symmetry. [166,167] Therein were observed not only the intriguing EPs, but also unprecedented phenomena not seen in Hermitian systems, such as unidirectional transports [168,169] and enhanced sensitivity. [170][171][172][173] The primary studies of PT-symmetry and exceptional physics deal with finite gain/loss entities coupled to each other. Via the manipulations of the gain/loss strength, the entity geometries, and the couplings, various EPs were demonstrated in parameter spaces, including both the normal second-order EPs of two-state coalescence [171,174] and the higher-order EPs of multiple-state coalescences. [172,175] Refer to Figure 19A,B for the schematic second-and third-order EPs, respectively, in the two-and three-level systems. It has been shown that in addition to the novel physics and potential applications aforementioned, the EPs also yield the nontrivial spectral topology, which can be identified by encircling these singular points in their eigenvalue Riemann surfaces. [176,177] As illustrated in Figure 19C, when encircling an EP once the eigenvalues swap. Only after another round, the eigenvalues return to their initial positions. In this process, the eigenstates acquire a π Berry phase, accounting for a fractional topological charge, which is a unique topological characteristic in the non-Hermitian systems. [177,178] Recently, the spectral topology of a third-order EP in a three-level system has also been exploited for its potential in realizing non-Abelian permutations of states. [179] For even higher N-level systems, it has been proven that through controlled loops, that is, the parameters are varied in the Riemann surfaces in controlled manners, eigenvalues are braided. [180] For N > 2, the braids of the eigenvalues in general form non-Abelian groups and therefore give rise to a variety of novel topological structures of knots and links in parameter spaces. [180]

| Non-Hermitian band topology in the momentum space
When considering the non-Hermitian ingredients in periodic arrays, complex band structures in the momentum space emerge and band topology comes into play, leading to both the novel gapless topological band degeneracies and the gapped non-Hermitian topological physics. In the following, we review these progresses.

| Gapless topology and exceptional degeneracies
Parallel to the developments of the gapless topological physics in Hermitian systems, non-Hermitian degeneracies have attracted increasing attention in recent years. In stark contrast to the band crossings in the Hermitian systems, the non-Hermitian degeneracies are typically EPs, exceptional lines, rings, and even links. Due to the simultaneously coalesced eigenvalues and eigenvectors at CHEN ET AL. | 203 these degeneracies, non-Hermitian systems enable interesting topological phenomena that are usually different from their Hermitian counterparts or even absent in the Hermitian systems. For example, it has been shown that in 3D Hermitian systems featuring band degeneracies like Weyl points, there exist open surface Fermi arcs connecting these degenerate points with opposite chirality (i.e., opposite topological charges), [182] as illustrated in Figure 20A. Not unique to such a Hermitian scenario, open Fermi arcs have been shown to also appear in non-Hermitian systems. [183] Zhou et al. [184] proposed a photonic crystal slab that radiated energy to the surrounding ( Figure 20B) and demonstrated that in such a dissipative system, open Fermi arcs emerged as consequences of EPs, that is, they connect a pair of EPs with opposite fractional topological charges (the fractional topological charges have the similar origin to the spectral topology identified in the parameter spaces as discussed above). This is different from the integer topology charges of the Weyl points. Such fractional (half) topological charges have been suggested to be promising for the generation of half-integer vector-vortex beams. It is also pointed out that unlike the manifestation as a surface property in the Hermitian Weyl systems, the open Fermi arcs in this non-Hermitian system have been found to reside in the bulk bands. [184] In addition to the EPs, the considerations of non-Hermitian ingredients in periodic arrays have been demonstrated to enable more versatile exceptional degeneracies. In Cerjan et al., [185] non-Hermitian effects were added to a Hermitian Weyl material and shown to be able to spread the Weyl point into a ring of EPs (see Figure 20C). Such a Weyl exceptional ring surprisingly maintains the real and quantized Berry (topological) charge and hence, there still exist surface Fermi arcs similar to the cases with Weyl points. Differently, the ring shape constitutes a novel topological object as the nonpoint source of the Berry flux, which can no longer be interpreted as a magnetic monopole of the Berry curvature (the latter is standard for the Weyl points). Beyond what has been experimentally realized, theories have predicted more possible non-Hermitian degenerate structures (see Figure 20D-F), such as the exceptional Hopf-links generated by adding non-Hermitian perturbations to a nodal line semimetal, [186] the non-Hermitian knotted nodal degeneracies without any fine-tuning or symmetries, [187] symmetry-protected exceptional nodal points and chains, [188,189] and so forth. Associated with these intriguing exceptional degeneracies, curious transport effects have also been reported. For example, unlike its Hermitian counterpart supporting 1D nodal lines, the Hopf-link exceptional line semimetal hosts non-Hermitian Fermi arcs that are 2D twisting surfaces with Hopf-link boundaries, [186] as depicted by the green shading in Figure 20D.

| Non-Hermitian complex bandgaps
In addition to the gapless non-Hermitian topological features, a great proportion of studies have been devoted to the gapped non-Hermitian topological phenomena. As discussed in previous sections, the gapped topological phases in Hermitian systems are usually referred to as F I G U R E 19 EPs in the parameter spaces and their spectral topology. (A, B) Respectively, the second-order and third-order EPs in the two-and three-level systems. Reproduced with permission. [181] Copyright 2022, Taylor & Francis. For the sake of clarity, only one parameter (i.e., the gain/loss contrast) is varied. (C) The spectral topological properties of a second-order EP illustrated on their Riemann surfaces. Reproduced under terms of the CC-BY license. [178] Copyright 2018, The Authors, published by Springer Nature. EPs, exceptional points.
insulators, accompanying bandgaps in which the topological boundary states emerge. When extending to non-Hermitian scenarios, the definition of bandgaps becomes trickier. In Hermitian systems, the eigenvalues are pure real and permit bandgaps in which no states are allowed. In non-Hermitian systems, however, the eigenvalues are typically complex and it is unlikely to straightforwardly generalize the concept of Hermitian bandgaps.
Nevertheless, two possible ways have been proposed to define a complex bandgap. One is referred to as the point gap and the other as the line gap. [190] Specifically, a non-Hermitian Hamiltonian is said to have a point gap if there exists a base point (an energy/frequency point) , where H k ( ) represents the non-Hermitian Hamiltonian and k is the wave vector. From the above definition, one can directly identify a special instance where the non-Hermitian Hamiltonian has only one band but supports a point gap (see Figure 21A). This essentially corresponds to a unique non-Hermitian topological phenomenon that will be reviewed and discussed in detail in Section 5.3. Alternatively, the line gap refers to the situations where the non-Hermitian complex spectra are separated by 1D lines. Technically speaking, the line gaps can be of any offset and orientation with respect to the real and imaginary axes. However, by a constant energy/frequency shift and/or by rescaling the Hamiltonian with a complex constant (corresponding to the rotations of the energy spectra), the line gaps can be accordingly aligned with the imaginary or real axis, forming the so-called real or imaginary line gap, respectively (see Figure 21B,C). Although the shift and/or rotations may violate or transform the non-Hermitian symmetries, it has been found to be convenient to distinguish the real and imaginary line gaps based on their distinct topological behaviors. [191] In fact, based on the identifications of the point, real and imaginary line gaps, efforts have been made to classify the non-Hermitian band topology, [191][192][193] parallel to the celebrated AZ classifications in the Hermitian realm.  [185] Copyright 2019, Springer Nature. (B) A non-Hermitian photonic crystal slab that supports EPs. Each pair of EPs is connected by a bulk Fermi arc (the blue curve). Reproduced with permission. [184] Copyright 2018, The American Association for the Advancement of Science. (C) The exceptional rings formed by introducing non-Hermitian perturbations into a Hermitian system with Weyl points, where the Weyl points spread into exceptional rings. Reproduced with permission. [185] Copyright 2019, Springer Nature. (D-F) Various theoretical proposals on exceptional knots and links. Reproduced with permission. [186][187][188] Copyright 2019 American Physical Society. EPs, exceptional points.

| Gapped non-Hermitian topological phases
Typically, the non-Hermitian models are created by adding non-Hermitian perturbations to the Hermitian topological models. There are in general two sources of non-Hermiticity. One is by introducing on-site gain/loss and the other relies on asymmetric couplings. In some models, these two sources are combined. Adding on-site gain/loss usually produces non-Hermitian line gaps. It has been shown that the topological properties of the line gaps are essentially Hermitian in the sense that any line-gapped non-Hermitian bands can be continuously deformed into being either Hermitian or anti-Hermitian (the latter indicates Hamiltonians that can be changed to their Hermitian counterparts by simply multiplying the imaginary unit i). [190] Correspondingly, the non-Hermitian topological invariants can be determined by following the Hermitian schemes. Introducing asymmetric couplings, on the other hand, gives rise to genuine non-Hermitian topology that can fundamentally alter the Hermitian topological properties or is absolutely absent for Hermitian models, as will be discussed in Section 5.3.
Here, we first focus on the gain/loss-induced non-Hermitian topological phenomena. Therein, a widely studied model is the complex SSH model. [194] As shown in Figure 22A, such a model consists of the typical dimerized chain decorated with staggered on-site gain and loss (or they can be staggered on-site loss and loss for the sake of experimental feasibilities). Note that while the sublattice symmetry is lost, this model preserves the chiral symmetry. By choosing proper coupling and gain/loss parameters, a real line gap can be opened, which supports midgap edge states ( Figure 22A), just like that in the Hermitian SSH model. The winding number, similar to the Zak phase describing the Hermitian nontrivial band topology, can also be obtained for the non-Hermitian model. [195] In addition to these common topological phenomena, there are novel features emerging in the non-Hermitian model. For example, when the gain/loss contrast is moderate such that the system is in the PTsymmetric phase, the eigenvalues of the bulk states remain pure real as expected. However, the edge states carry net gain/loss and exhibit dynamic behaviors (see the inset of Figure 22A), that is, when considering transient behaviors, one edge state will be amplified over time while the other one will be diminished. [196] This potentially can be used in efficient mode selections where the gain edge mode can provide an enhanced visibility. [197] The above-discussed inconsistence between the eigenvalues of the bulk and edge states has also raised debates whether the PTsymmetric systems support topological interface states that also obey PT-symmetry. In a later work, Weimann et al. [198] proved that it is indeed possible to obtain PTsymmetric topological states, which emerge at the interface between two non-Hermitian SSH lattices with opposite gain/loss staggers.
Following the intriguing topological properties in the gain/loss-induced non-Hermitian systems, another branch develops quickly and explosively, that is, the topological lasing. Starting from the 1D non-Hermitian SSH model, the 0D edge states with enhanced visibility have been employed as stationary cavity modes and extensively explored for lasing. [199,205,206] It was shown that in favor of the topological nature, the edge mode lasing exhibited a high reliability and was rather robust to defects and disorders (see Figure 22B). [199] Accompanying with the rapid developments of various topological phases of matter in 2D Hermitian systems, versatile topological lasing has been also explored by incorporating gain/loss into these systems, as schematically illustrated in Figure 22C-F. Bahari et al. [200] proposed topological cavities with arbitrary geometries, which supported unidirectional topological edge states owing to the famous QHE. They showed that by pumping the cavities equipped within gain materials, topological lasing with arbitrary geometries occurred. It was further demonstrated that such a topological lasing can also be coupled to selected waveguide outputs with an isolation ratio in excess of 10 decibels. More recently, a verticalcavity laser has been reported in the topological crystalline insulators whose topological nature came from compressing or stretching the honeycomb lattices. [201] The lattices were made of arrays of emitters (or lasers). The topological edge states emerged at the interface between the compressed and stretched lattices. It was shown that the edge states can force the injection locking on all the interface lasers, which correspondingly led to a coherent single lasing and provided a promising technique for large-scale lasing. To obtain a single-mode laser, Yang et al. [202] proposed that the Dirac-vortex topological cavity is also a possible candidate. They showed that the 2D topological defect modes in the Dirac-vortex cavity offered an optimal selection for the single-mode lasing in two dimensions, which might surpass the standard industrial products of distributed feedback lasers and vertical-cavity surface-emitting lasers with respect to both the lasing power and the divergence angle. In the acoustic community, attempts to realize sasers (acoustic versions of lasers) have been also made. For example, Hu et al. [203] used the electrothermoacoustic coupling to supply an acoustic analog of gain medium and realized what the authors referred to as the "audio lasing" modes.
In addition to act as lasing materials, gain and loss have also been proven to be able to directly control the topological properties and give rise to TPTs. Zhao et al. [204] showed that via selective pumping on a topological array decorated with gain and loss materials, TPTs happened and the associated edge states emerged on the boundary of  [199] Copyright 2018, The Authors, published by Springer Nature. (C-F) Respectively, the topological cavity laser with arbitrary geometries (Reproduced with permission, [200] Copyright 2017, The American Association for the Advancement of Science), the vertical-cavity laser based on the honeycomb lattices (Reproduced with permission, [201] Copyright 2021, The American Association for the Advancement of Science), the Dirac-vortex topological cavity laser (Reproduced with permission, [202] Copyright 2022, Springer Nature) and an acoustic "audio lasing" based on the triangular lattices (Reproduced with permission, [203] Copyright 2021, Springer Nature). (G) A topological router enabled by optical-pumping on a topological array decorated with gain and loss media. Reproduced with permission. [204] Copyright 2019, The American Association for the Advancement of Science. the pumping area, leading to the dubbed topological routing effect ( Figure 22G). Gao et al. [207] demonstrated that solely by introducing non-Hermiticity (implemented by loss materials), topological edge states emerged in an acoustic system consisting of arrays of cavities, while in the Hermitian limit, the system was gapless. The very same authors also implemented this mechanism into the HOTIs and realized higher-order topological corner states solely by introducing non-Hermiticity. [148] Additional to the photonic and acoustic realms, exploring non-Hermitian factors as an indigent for novel topological phases has been made in electric circuits, where the positive and negative resistors in circuits were tuned to obtain controlled gain and loss and therefore the TPTs. [208] It was also showed that by further tuning the resistors, one can even switch between different topological phases, demonstrating high versatilities.

| Non-Hermitian topological physics beyond the Bloch scope
In Section 5.2, we have reviewed the developments of topological physics in on-site gain/loss-induced non-Hermitian systems. Next, we turn to a recently emerging branch that has attracted tremendous attentions due to its unique non-Hermitian topological phenomena (or often mentioned as genuine non-Hermitian topological phenomena). These intriguing phenomena are enabled by asymmetric couplings. It is known that the non-Hermiticity of a Hamiltonian comes with various forms. For the on-site gain/loss, the non-Hermiticity originates from the complex diagonal elements. There is another form of non-Hermiticity, that is, the distinctions between the off-diagonal elements. Reflected on the physical lattices, these distinctions correspond to the asymmetric couplings. This type of non-Hermiticity has been shown to bring forward interesting non-Hermitian topological physics, such as nontrivial band topology for single-band systems, breakdown of conventional BBC, non-Bloch PT symmetries, and so forth. In the following, we review these progresses.
As mentioned above, in non-Hermitian systems, it is possible to have one band but support a point gap. The prototypical model is known as the Hatano-Nelson (HN) model, [209,210] which is a 1D disordered tight-binding lattice with nearest-neighbor asymmetric couplings (i.e., the left coupling strength is different from the right one). This model was originally proposed to demonstrate that the nonreciprocity (originated from the asymmetric couplings) can prevent Anderson localizations. Recently, it has been revisited for its unique unidirectional wave dynamics. Specifically, consider a clean HN model without disorders and on-site potentials such that the only non-Hermitian source is the asymmetric couplings ( Figure 23A). The energy band of this model is shown in Figure 23B. It is seen that in this system, the eigenenergy E becomes complex, whose real part is symmetric with respect to the wave vector k. The imaginary part, on the other hand, exhibits an asymmetric property, which precisely indicates the nonreciprocal nature of the HN model. Such an asymmetric property suggests an asymmetric transport behavior, that is, the probability of the left hopping does not equal to that of the right hopping. When encountering with boundaries, the unequal hopping leads to wave localizations favorable to the direction with stronger hopping (in the present example, it is the left direction). To see this, the eigenmode profiles for a finite HN chain with both left and right open boundary conditions (OBCs) are presented in Figure 23C. It is shown that indeed, all the eigenmodes are localized toward the left boundary, exhibiting a fundamentally different behavior from the Hermitian finite lattice that typically supports Bloch oscillations when terminated by open boundaries. Such a non-Hermitian localization effect shares a phenomenological similarity to the skin effect in electric conductors, and therefore is dubbed NHSE and the localized modes are referred to as the skin modes. [181,211] It has been suggested that the NHSE has a topological origin. Re-examine the eigenenergies of the HN model in the complex E Re( )-E Im( ) plane and note that they form a closed loop (see Figure 23D). The arrow denotes the wave vector k varying from −π to π. If one takes a based point E B inside the loop, it constitutes a point gap, which resembles the one shown in Figure 21A. By phasewinding the eigenenergies, it encircles E B one time, giving rise to a nontrivial winding number of 1 (taking the convention that anticlockwise encircling imprints positive winding numbers). Such a nontrivial phasewinding of the eigenenergies rather than the eigenvectors is a unique topological feature in the non-Hermitian systems owing to the asymmetric couplings and is elusive in the Hermitian systems. The topological interpretation is elucidated in the following. According to the conventional BBC, for each base point E B , a nonzero winding number 1 indicates one edge state with energy E B . Since there can be an infinite number of E B inside the loop, this indicates infinite edge states. These edge states are manifested at the boundary of a semi-infinite non-Hermitian chain. [212] Upon full OBCs, due to the finite length, the chain is no longer capable of accommodating infinite edge states. Nevertheless, the spectrum under the full OBCs is still embedded in the spectrum under the semi-infinite condition, as the latter case is an extrapolation of the former case under L → ∞, where L denotes the chain length. [212] This indicates that the eigenstates under the full OBCs still preserve the behaviors as the edge states in the semi-infinite chain, that is, exponentially localized toward boundaries, corresponding to the non-Hermitian skin modes as identified in Figure 23C.
The NHSE, with the intriguing localized skin modes, has fundamentally changed the understanding on the topological band theory and enabled novel physics and interesting phenomena beyond the Bloch scope, hence igniting a broad research, including the theoretical interpretations, [213,214] the experimental demonstrations, [215][216][217] and the extensions to higherdimensions. [218] In particularly, it has been found that when interplayed with conventional topological phases which rely on symmetries, the asymmetric-couplinginduced non-Hermiticity can profoundly alter the topological properties of finite lattices with OBCs, leading to the breakdown of the conventional BBC. [211] The conventional BBC dictates that the nontrivial band topology identified under the PBCs is able to predict the emergence of topological boundary states under OBCs and vice versa. Take the example of the Hermitian SSH model ( Figure 24A) and plot the eigenspectra under PBCs ( Figure 24B) and OBCs ( Figure 24C). It is seen that under the PBCs, there exists a bandgap-opening-closing-reopening process, signaling a TPT. In the topologically nontrivial phase, the SSH chain supports topological edge states, which are manifested under the OBCs (see the red curves in Figure 24C). These edge states are distributed at the two ends of the finite SSH chain and exhibit typical localization behaviors, as shown in Figure 24D for three exemplified cases. When introducing asymmetric couplings ( Figure 24E), however, bandgap closing point identified under PBCs can no longer predict the emergence of the topological edge states under OBCs, as indicated in Figure 24F,G. Such a deviation calls for re-examinations of the conventional BBC.
Targeting this issue, a generalized Brillouin zone (GBZ) method in the scope of non-Bloch band theory has been proposed to establish the non-Hermitian BBC. [211,216,220] Roughly speaking, this theory considers the wave dynamics in the non-Hermitian systems to be non-Bloch and governed by a complex wave vector k k i r r ′ = − ln , ∈ . The GBZ is defined based on k′ and the new phase factor of the non-Bloch waves yields β e re = = ik ik ′ . Compared with the conventional BZ with the phase factor e ik that encloses a unit circle as k varies from π − to π, the GBZ with β in general does not equal to the unit circle as r typically deviates from 1, leading to the non-Bloch transports with directional amplifications or attenuations. This is consistent with the NHSE features discussed above. Using the GBZ method, Yao and Wang [211] successfully derived the TPT for the non-Hermitian SSH model and found that the transition points were indeed quantitively different from the energy gap-closing point identified under PBCs, providing concrete evidence to the breakdown of the conventional BBC in asymmetric-coupling-induced non-Hermitian systems. Subsequently, they evaluated the non-Hermitian topological invariant defined on and run over the GBZ, which faithfully determined the non-Hermitian topological edge states under OBCs and hence established the non-Hermitian BBC, reminiscent of the conventional BBC where the topological invariants evaluated over the BZ faithfully determine the emergence of the Hermitian edge states under OBCs. It is pointed out that the asymmetric couplings not only alter the Hermitian BBC, but also have a nontrivial effect on the topological edge states. As shown in Figure 24H, driven by the non-Hermiticity, the edge states become skin modes when the non-Hermitian strength is large enough, while for weak non-Hermiticity, the edge states remain separately localized at the two chain ends, similar to their Hermitian counterparts (see Figure 24D). This phenomenon has been identified as the effect of the NHSE competing with the crystalline symmetries. [219] The crystalline symmetries usually protect the Hermitian band topology and their interactions with the NHSE promise rich non-Hermitian physics. Following this direction, a fruitful research has been conducted to understand the non-Bloch band theory and to propose novel non-Bloch topological phenomena. [221][222][223] In addition to the band physics, asymmetriccoupling-induced non-Hermiticity has also contributed to the novel exceptional features. In Martinez Alvarez et al., [213] it has been shown that in the non-Hermitian systems with asymmetric couplings, a new type of EP emerges, dubbed the non-Bloch EPs. [224] Unlike the EPs induced by on-site gain/loss, which are rather sensitive to the parameter changes, the non-Bloch EPs can pervade over a wide parameter regime and are possibly associated with the emergence of NHSE. [213,225] These EPs have been further investigated and found to be the unique properties under OBCs. [224] When generalizing to two and higher dimensions, it has been shown that even more surprisingly, only by increasing the dimensions, the non-Bloch PT phase transition universally approaches to zero as the size of the non-Hermitian system increases, whereas the Bloch PT phase transition and the 1D non-Bloch PT phase transition typically have nonzero thresholds. [226] This suggests rich and unexpected interplay among PT symmetries, exceptional features, NHSE, and spatial dimensions, holding great promises to versatile non-Hermitian physics and phenomena. insulators with TRS invariant. The former was proposed very early, but requires external fields, such as magnetic field, rotating medium, and airflow, which limits its applications. The latter releases the requirement of external fields, but suffers from the mini-gap, which weakens its robustness. Researchers try to find a way to break TRS without external fields. Recalling the role of space periodicity in the band theory, the introduction of time periodicity into the system may solve this problem, which leads to the concept of FTI.
Floquet theory is a mathematic theorem proposed by Floquet in the 19th century, which gives a general method to solve differential equations with periods. A well-known application is the Bloch theory in solid-state physics with space periodicity. In a periodical lattice, the wavefunction is the Bloch wave with the form: If we replace time periodicity with space, the solution of the time-dependent evolution Hamiltonian will also exhibit energy band structures, similar to Bloch bands, following the Floquet theorem.
In photonics or phononics, the time-dependent state evolution is challenging since the lifetime of states is related to the quality factor and the state evolution follows the wave equation instead of the Schrodinger equation. In 2013, Mikael C. Rechtsman et al. wisely used the z-axis to replace the role of time by mapping the Schrodinger equation to the Paraxial evolution equation, and realized photonic FTI for the first time. [227] In their experiments, waveguides assemble hexagonal lattices in the xy-plane but extend to the zdirection, along which performs the topological protected propagating states. This groundbreaking work opens the gate for FTI not only in photonic but also in acoustic systems. In 2014, Michael Pasek et al. show that simple 2D coupling ring waveguides (CRWs) can be taken as FTIs. [228] Their solid theoretical demonstration reveals the intrinsic connection between FTI and existing optical CRWs. Thus, we summarize three main approaches to realize FTI based on the timedependent modulation (t-modulation), z-dependent modulation (z-modulation), and CRW model, respectively.
It was shown that FTIs not only benefit from the topology-protected chiral states as expected, but also show special properties that have never been found in other topological systems. For example, topological edge states with zero bulk Chern number, namely, anomalous FTI (AFTI). [229] On the other hand, the TP, in which a prominent example is Thouless pumping, [230] can be regarded as a special case of FTI with an infinite time period. Therefore, we will review several progressed acoustic TP as a supplement to FTI in this chapter.
This section is organized as follows. We briefly review the basic concepts of the FTI in Section 6.2. In Section 6.3, the developments of FTI will be discussed in detail based on three main categories, t-modulation, z-modulation, and the CRW model. The recent development about higher-order FTI is also included in this part. In Section 6.4, TP will be shown briefly. Finally comes the conclusion.

| Basic concepts of the FTI
In a Floquet system, the Hamiltonian is periodically modulated in time: in which T is the period. During an extremely short After one period we can get (13) in which U T ( ) is called the evolution operator. On the other hand, Floquet theory indicates that: Equations (13) and (14) imply in which ε is a real number. It implies that we can define an effective Hamiltonian: In this new Hamiltonian equation, ε plays the role of eigenenergy, for which we call it quasienergy. Reviewing this calculation process, we find that U T ( ) covers all the evolution processes during one period. As a result, "quasi energy" ε, to some extent, reflects the average of the Hamiltonian of the system in this period, and has a π T 2 / periodic. Therefore, we can get the information about the system from this quasienergy band. We can redefine the Chern number of quasienergy bands following that for static systems: It is worth noting that, in Floquet systems, one can find the case that the edge states appear even in a system with zero bulk Chern number, and gives a classification of the AFTI. In Chern insulators, the number of edge states is associated with the gap Chern number, which is defined by summing all the bulk Chern numbers below the gap. In contrast, in AFTI, the gap Chern number is not well defined since the system exhibits infinite energy bands with π 2 periodic. While the existence of edge stages is consistent to the bulk-edge correspondence, since we can define the bulk Chern number as the number difference of edge states in the adjacent gaps below and above that bulk band.
We can redefine a new topological invariant "winding number" (n edge ) to describe AFTI systems. Evolution operator U t ( ) can be abbreviated as U , and n edge can be written as One can relate C of a bulk band with two n edge of adjacent gaps n n C − = .

| Acoustic FTI based on t-modulation
In 2016, Romain Fleury et al. proposed a 2D acoustic FTI by periodically compressing the volume of fluid materials. [231] As shown in Figure 25A, they utilize silicone rubber RTV-602, an ultralow loss material, to construct the crystal system. The effective acoustic capacitance C βV = can be controlled by compressing the volume C t δC ω t φ Δ ( ) = cos( − ) m m m . By monotonically increasing or decreasing the phase φ m of three sites in one unit cell, the system turns to topologically nontrivial, that is, one can observe unidirectional acoustic propagating on the edge. This proposal was finally experimentally realized in 2020. [233] They used an array of shunted piezoelectric disks bonded to a thin polylactic acid plate, while modulating them by a periodic on-site potential. The transmission experiment proved the existence of robust edge states. Unsuspectingly, the design of the t-modulation platform provides new ideas for metamaterials. In 2018, Theodoros T. Koutserimpas et al. found that the FTI system presents the zero-refractive phenomena. [234] In 2019, an aeroacoustics t-modulation system was proposed. [235] The authors planned to modulate the airflow by temperature. However, it is much more difficult to regulate airflow than fluid, and this proposal remains at the theoretical stage till now.

| CRW acoustic FTI (including AFTI)
To realize FTI through t-modulation is experimentally challenging. An alternative and experimental feasible approach is the CRW model. It can be proved that the transformation matrix of a 2D CRW model equals to evolution operator of one period in the t-modulation system, which makes the CRW method equivalent to realize FTI. [228] In 2016, Yu-Gui Peng et al. constructed an AFTI in an acoustic CRW model. They used acoustic metamaterial waveguides to construct air-metal layers periodically stacking along the ring as shown in Figure 25B. [232] The acoustic wave propagates along the ring clockwise or anticlockwise, composing a timereversal symmetric pair. Following the theoretical analysis in Rudner et al. [229] the system is an AFTI, and one can observe gapless edge states despite the zero-Chern number bulk band. This work was the first realization of the acoustic AFTI, which paved the way for further studies. However, due to the thermal viscosity and dispersive coupling strength between unit cells, this design suffers from significant loss and narrow band. In 2017, the model was further optimized, and around 84% in that frequency range is predicted to be filled with topological bands. [236] A similar system was proposed by Wei et al. in 2017. [237] They used a simpler structure to realize acoustic AFTI with high transmissivity.

| z-Modulation FTI
In photonics, Rechtsman gave a good example to use the z-axis to replace the t-axis. However, since the tremendous impedance mismatch between waveguides and background air, the acoustic wave is confined in the waveguides and no evanescent wave exists to couple adjoint waveguides. Thus, additional waveguides in acoustics are used to couple the waveguide modes. In 2018, Yu-Gui Peng et al. followed this way and theoretically proposed a 1 + 1-dimensional acoustic FTI. [238] Their model can be taken as the wellknown SSH model along the x-direction, and has alternative couplings along the z-direction. Their results showed the phase transition via tuning the coupling strength and topologically protected interface states. They further extended their design to 2 + 1 dimension, and experimentally realized a 2 + 1-dimensional acoustic FTI. [239] As shown in Figure 26A, they utilized the judicious approach to attach the crewed coupling arms to waveguides in the experiment and largely improved the coupling efficiency. The topological property is confirmed by the existence of paired Weyl points and Fermi arc-like surface states.

| Higher-order FTI
Very recently, higher-order (HO) acoustic FTI has been realized in a periodically driven bipartite square lattice model hosting multiple topological phases. [240] As shown in Figure 26B, an eight-stepped Hamiltonian was used, and the HO phase can be observed along a few critical lines in the phase diagram. In 2022, this idea was realized by the z-modulation method as shown in Figure 26C. [147] Interestingly, they found the oscillation of the corner localization, a new phenomenon that has never been found in static HOTI. Their finding also extends the definition of AFTI, because their quasienergy bands in the HO phase have both zero-quadrupole moment and zero-polarization, the trivial topology mark in static HOTIs instead.

| Acoustic TP
The topology is not only reflected in the quantized number of boundary states, but also on the quantized transportation of charges. Such quantized transportation can be realized through adiabatic evolution, being the quasistatic limit of t-modulation in FTI. Thouless pump, proposed by Thouless in 1983, shows the transport of a precisely quantized amount of charge during an adiabatic cyclic variation of the lattice potential in a 1D lattice. It is regarded as a dynamical version of the 2D QHE. In 2012, Yaacov E. Kraus et al. realized the first adiabatic Thouless pumping in photonic quasicrystals. [241] They arranged waveguides extending on the z-axis and constitute 1D Aubry-Andre-Harper (AAH) model in the other direction. The phase parameter continuously changes along the z-axis and plays the role of a F I G U R E 25 Acoustic FTI. (A) Acoustic FTI based on the t-modulation method. The material is silicone rubber, a water-like ultrasonic material. There are three interconnected cylindrical cavities in one unit cell, in which the capacitance is periodically modulated. Reproduced under terms of the CC-BY license. [231] Copyright 2016, The Authors, published by Springer Nature. (B) Acoustic FTI based on the CRW method. The sample is composed of 2D coupled metamaterial rings. One can see the simulation and experimental results of the one-way edge states. Reproduced under terms of the CC-BY license. [232] Copyright 2016, The Authors, published by Springer Nature. 2D, two-dimensional; CRW, coupling ring waveguide; FTI, Floquet topological insulator. synthetic dimension wavevector, which makes the 1D quasicrystal system exhibit topological properties that were thought to be limited to 2D systems. This work inspired the realization of acoustic TP, and there have been growing pieces of literature investigating acoustic TP using additional DOFs to replace time.
In 2019, Ya-Xi Shen et al. proposed and experimentally demonstrated the unidirectional sound localization in 1D waveguides. [242] As shown in Figure 27B, sound energy can be directly transferred from an excited cavity to the target cavity but restricted in the opposite direction. In the same year, Xiang Ni et al. used a series of 1D acoustic AAH models with different parameters to construct topological edge states as shown in Figure 27C. [243] With elegant adjustments of experimental parameters, they observed the Hofstadter butterfly and fractal edge spectra. However, it was realized in a discrete space of parameters. In 2020, Emanuele Riva et al. presented the first experimental demonstration of TP in continuous elastic plates, extending this topic to elastic wave systems. [244] In 2021, Ze-Guo Chen et al. demonstrated the dynamic transfer of topological states in acoustic waveguides. [245] Utilizing the finite size induced mini-gap of edge states dispersion and adjusting the rate of parameter evolution, they successfully observed a nonadiabatic transition that follows the Landau-Zener model. Meanwhile, Hui Chen et al. used a similar way to introduce synthetic dimension and evidenced the topological sound transport through edge-to-edge topological and corner-to-corner TP associated with the 2D and 4D QHE. [146] In 2022, Zhaoxian Chen et al. modulated the coupling of the AAH model, but not on-site energy as that in previous work, and observed acoustic TP and Hofstadter's butterfly in simulation. [247] On the other hand, t-modulation is challenging as it requires fast modulation to overcome the dissipation of the state before it is pumped from one edge to another. In 2020, Wenting Cheng et al. overcame the difficulty and realized acoustic TP based on fast modulation of structures. [246] F I G U R E 26 HOFTI based on the z-modulation method. (A) A schematic of acoustic FTI with stepped modulated couplings along the zdirection. Reproduced under terms of the CC-BY license. [239] Copyright 2019, The Authors, published by the American Physical Society. (B) Phase diagram of the HOFTI. Reproduced with permission. [240] Copyright 2021, American Physical Society. (C) Floquet corner states with double the period of the underlying drive. Reproduced under terms of the CC-BY license. [147] Copyright 2022, The Authors, published by Springer Nature. FTI, Floquet topological insulator; HOFTI, higher-order FTI.

| Mechanical and elastic Chern insulators
As mentioned earlier, nonzero Chern numbers can only be obtained by breaking the TRS of crystals or artificial photonic/phononic crystals. For electronic or photonic systems, the TRS can be easily broken by applying an external magnetic field. However, phononic systems are immune to magnetic fields. This makes it particularly difficult to implement mechanical Chern insulators. An effective solution comes from the observation and understanding of the Aharonov-Bohm effect. It is known that when a charged particle bypasses the nonmagnetic field outside the solenoid, the magnetic field in the solenoid can still affect the phase of the particle. Also, the phase increases or decreases depending on which side of the solenoid the charged particles pass through. This physical mechanism of generating magnetic effects in spaces where there is no magnetic field is extremely valuable, for example, designing Chern insulators for phonons. By engineering specific hopping amplitudes between the lattice sites of the artificial phononic crystal, the total phase experienced by phonons after passing through a closed loop can be made nonzero. In this case, even in the absence of an external magnetic field, the TRS is also broken. This is demonstrated by the Haldane model, a tight-binding model famous for breaking TRS in the case of zero mean magnetic field. [248] F I G U R E 27 Acoustic TP. (A) The first example of z-modulated optical TP inspiring the development of acoustic TP. Reproduced with permission. [241] Copyright 2012, American Physical Society. (B) The coupling actions between neighboring cavities as the performance of one-way localized acoustic adiabatic pumping. Reproduced with permission. [242] Copyright 2019, American Physical Society. (C) Schematics of the edge state transfer during the parameter pumping with discrete samples. Reproduced under terms of the CC-BY license. [243] Copyright 2019, The Authors, published by Springer Nature. (D) Schematic of the acoustic TP in a consecutive model. The modulation range is shown in the red section. Reproduced with permission. [245] Copyright 2021, American Physical Society. (E) Schematic of the 3D channel modulated acoustic TP model. Reproduced under terms of the CC-BY license. [146] Copyright 2021, The Authors, published by Springer Nature. (F) Schematic of a pumping cycle modulated in time. A speaker and a microphone are inserted. Reproduced with permission. [246] Copyright 2020, American Physical Society. TP, topological pumping.
For mechanical and elastic waves, the first Chern insulator was reported by Nash et al. in 2015 using an electrically actuated gyroscopic metamaterial. [249] The metamaterial consists of dozens of coplanar gyroscopes, coupled adjacent to each other by small magnets placed inside each gyroscope. The precession of gyroscopes breaks the TRS of this mechanical metamaterial. Experiments show that there are chiral edge modes on the boundary of this dynamic metamaterial, that is, unidirectional and defect-immune transport of mechanical vibrations. It is worth mentioning that later in 2015, a similar gyroscope-based elastic wave Chern insulator was independently theoretically proposed. [250] A few years later, the same research team further showed that their system can be extended to a complex band topology by introducing tunable lattices, [251] and these gyroscopes still constitute Chern insulators despite in disorder, achieving the first amorphous TIs of any system. [252] Another idea for implementing Chern insulators in coupled mechanical systems, in a way similar to gyroscopes, is by means of the Coriolis force. [253,254] For continuous media, also in 2015, Peano et al. proposed the first elastic Chern insulator and topological elastic waves in an optomechanical system, [255] built on a well-designed Kagome phononic crystal with micrometer-scale features. When the entire phononic crystal is illuminated by the superposition of three laser beams, the laser beams form optical vortices and impart additional orbital angular momentum to the mechanical waves through optomechanical interactions. [256] By designing a suitable phase difference for the three laser beams, an elastic Chern insulator can be induced, the edges of which support unidirectional and antireflection elastic wave transmission. A major feature of this design is the capability of in situ modulation, which has been preliminarily verified by the latest experiments by Ren et al. [257] Another approach using optomechanical systems uses arrays of microtoroids in which adjacent microtoroids are coupled via evanescent radiation without direct mechanical coupling. [258] This scheme draws on the coupled resonator optical waveguides in photonics, under Floquet formalism. [259] Recently, the first elastic wave Chern insulator was experimentally demonstrated based on synthetic angular momentum bias in the form of coupled piezoelectric discs manipulated by external circuits. [233] In short, for Chern insulators for mechanical vibrations and elastic waves, there are solutions through timedependent elastic modulation, gyroscopes, optomechanical interactions, and Coriolis forces. And the first two have been implemented experimentally (Figures 28 and 29).

| Mechanical and elastic TIs under TRS
To mimic the TIs in mechanical systems, artificial Kramers pairs are needed, referred to as pseudospins ±½. One way to realize these pseudospins ±½ is through polarization. For all mechanical waves, Suesstrunk and Huber realized for the first TI by employing an array of macroscopic pendula, each with only one DOF of motion, but paired with multiple mechanical springs. [260] The Kramers pairs are formed by the polarization of the coupled 1D pendula. The pseudospins ±½ can be viewed as specific relative motion patterns of the two pendulums at each lattice site, which would correspond to left-and right-handed circular polarization in electromagnetism. The system exhibits mechanical helical edge modes that are topologically protected and thus immune to imperfections. Different from mechanical/elastic Chern insulators, since time-dependent modulation of the system is not required to break the TRS, mechanical TIs are much easier to implement for solid-state continuous elastic systems, and have more application value, especially for high frequencies and on-chip applications. Currently, various schemes have been proposed to simulate spins in elastic systems to construct TIs. Below, we will focus on introducing the most representative works related to the QSHs and QVHs.

| Elastic analogous of QSHEs
For analog quantum phenomena in two dimensions, elastic plate systems (especially the Lamb wave system) are an ideal playground. Mousavi et al. proposed the first elastic analog QSH in 2015, using a dual-scale phononic metamaterial. [261] Double Dirac cones are formed through symmetric (S-) or antisymmetric (A-) modes of plate elastic waves. Partial truncation of the bulk crystal without causing coupling of these two spin states creates further elastic helical edge states. Inspired by this work, helical edge states can also be generated in an elastic plate with patterned triangular holes, which has the through-thethickness symmetry broken by the drilling of blind holes on one side. [262] Besides polarization, modal hybridization is another effective way to obtain double Dirac cones. One simple way is through zone-folding techniques. As mentioned before, it was first proposed by Wu and Hu in 2015 for photonic crystals, [66] as an alternative symmetry-based approach to engineer an effective Dirac Hamiltonian with a tunable mass. In this way, the mass term is obtained by breaking the original translational symmetry without breaking the C 3 symmetry, creating a smaller BZ with original bands folded back. This allows all systems with DPs to obtain quadruple degeneracy with pseudospins. Utilizing zone-folding, topological transport of elastic waves was proposed by Chaunsali et al. on a plate phononic crystal with mounted resonators, [271] by Brendel et al. with snowflake patterned holes, [264] and by. It also successfully brought about the first on-chip phononic TI, in a nanoelectromechanical metamaterial made out of piezoelectric SiN to perform electromechanical transducing. [265] The first topological transport for surface acoustic waves (SAWs) also uses the zone-folding scheme, on a piezoelectric LiNbO 3 substrate with mounted microresonators. [272] Apart from zone-folding to obtain double Dirac cones, another representative strategy using the accidental degeneracy in the crystals. It takes advantage of the fact that C 6 crystals automatically features twofold degenerate symmetric states, p± and d±. If the crystals are carefully designed with a pair of p and d states at the same frequency, then there will be a double Dirac cone at the Γ point. The approach by accidental degeneracy of pairs of p and d led to the first experimental observation of elastic topological transport. [263] At the boundaries of two plate elastic insulators, elastic pseudospins transport locked with momentum are verified robust against various defects and distortions. Recently, elastic analogous of QSH and helical edge states are also implemented on an elastic rod lattice [273] and a plate-like TI with dislocation interface. [274] 7.2.2 | Elastic analogous of QVHEs As mentioned above, by breaking the space-inversion symmetry (SIS), the Dirac degeneracy is lifted which opens a topological bandgap and produces topologically protected valley edge states. This concept, referred to as the QVHE, can also be applied to elastic systems. The earliest approaches to elastic QVHE use resonators to produce opposite polarizations. Pal et al. proposed a plate platform with resonators a hexagonal lattice [275] and the same group experimentally verified the elastic F I G U R E 28 Mechanical and elastic Chern insulators. (A) Honeycomb lattices with and without dynamic spin, phononic bands of the later have with nonzero Chern numbers. (B) Mechanical chiral edges states. Reproduced with permission. [250] Copyright 2015, American Physical Society. (C) Experimental gyroscope, coupled with adjacent by magnetic force, forming the first mechanical Chern insulator with chiral edge states. Reproduced with permission. [249] Copyright 2015, National Academy of Sciences of the United States of America. (D) First amorphous TI. Reproduced with permission. [252] Copyright 2018, Springer Nature. (E) 2D non-mass-spring system in a rotating frame, imposed by Coriolis force. Reproduced with permission. [254] Copyright 2015, IOP Publishing. (F) proposed topological optomechanical nanobeam lattice. Reproduced with permission. [258] Copyright 2020, Springer Nature. (G) First Chern insulator for elastic waves. Reproduced under terms of the CC-BY license. [233] Copyright 2020, The Authors, published by The American Association for the Advancement of Science. 2D, two-dimensional; TI, topological insulator. topological valley modes. [266] Similar schemes with the help of resonators have also been experimentally confirmed several times in the follow-up, for example, in a diatomic-graphene-like lattice [276] and localresonate plate phononic crystal with multiple operating frequencies. [277] In elastic materials, the strain field is another effective way to produce valley edge states. For example, the SIS of a truss-like hexagonal lattice can be broken by introducing uniform elastic deformations of the lattice. [278] As a consequence, edge states for flexural waves (A-modes of the Lamb wave) at the interface of two spatially inverted lattices can be produced, and the topological transition can occur by simply tuning the pressure within the units.
Furthermore, the edge states can be excited independently in a single lattice configuration by setting proper boundary conditions. Since the QVH only requires only a simple breaking SIS, its design is easiest compared with QHE and QSH. Therefore, this scheme is advantageous for on-chip integration. In 2018, Yan et al. successfully realized elastic valley edge transport on silicon chips patterned with arrays of triangular pillars. [279] Valley pseudospin is achieved through convolution of each triangular pillar around its vertical central axis during vibration. After that, elastic valleylocked edge states have been demonstrated in nanoelectromechanical suspended SiN membranes [267,268] and nanoscale optomechanical crystal. [257] With the F I G U R E 29 Mechanical and elastic topological insulators (TI). (A) First mechanical TI is composed of coupled pendula. Reproduced with permission. [260] Copyright 2015, The American Association for the Advancement of Science. (B) First proposal for elastic TI using a dual-scale phononic metamaterial. Reproduced under terms of the CC-BY license. [261] Copyright 2015, The Authors, published by Springer Nature. (C, D) First two elastic TIs, both achieved on a patterned elastic plate. Reproduced under terms of the CC-BY license. [262] Copyright 2018, The Authors, published by American Physical Society. Reproduced with permission. [263] Copyright 2018, American Physical Society. (E) Arbitrarily shaped elastic topological waveguide at the nanoscale, simulated using a tight-binding model. Reproduced with permission. [264] Copyright 2018, American Physical Society. (F) First on-chip phononic TI. Reproduced with permission. [265] Copyright 2018, Springer Nature. (G) Elastic analog of the QVH with valley-protected edge state. Reproduced with permission. [266] Copyright 2017, American Physical Society. (H, I) On-chip valley phononic crystals based on Si and suspended SiN membranes. Reproduced with permission. [267] Copyright 2021, John Wiley and Sons. Reproduced under terms of the CC-BY license. [268] Copyright 2021, The Authors, published by The American Association for the Advancement of Science. Reproduced under terms of the CC-BY license. [257] Copyright 2022, The Authors, published by Springer Nature. (J) First GHz phononic TI architected on AlN thin film. Reproduced with permission. [269] Copyright 2022, Springer Nature. (K) On-chip valley-protected SAWs, with extended apertures for idea matching to interdigital transducers. Reproduced under terms of the CC-BY license. [270] Copyright 2022, The Authors, published by Springer Nature. SAW, surface acoustic wave. development of integrated phononics in recent years, we highlight two representative works in the following. Zhang et al. first realized a topological phononic waveguide at 1.06 GHz using a microscale AlN snowflake phononic crystal. [269] It marks the breakthrough of topological mechanical transmission over GHz, the desired frequency for modern mobile communication and quantum information processing. Wang et al., proposed and experimentally demonstrated an extended valley-locked SAWs in a topological heterostructure [270] on LiNbO 3 substrate. It provides a critical approach for idea matching between phononic waveguides and electromechanical transducers for phonon excitation and reception.

| TOPOLOGICAL PHASES IN QUASISTATIC ELASTIC MECHANICAL SYSTEMS
The previous sections reviewed the topological phases in acoustic metamaterials, which involves the propagation of waves in air and solid. This part focuses on topological phases in quasistatic elastic mechanical systems. We first introduce the topological phases in Maxwell lattices. If the d-dimensional frame composed of N b bars and N j nodes is stable, then Here z is the average coordination number, representing the number of links per node. This criterion is also known as Maxwell's criterion, and structures that satisfy this criterion are often referred to as Maxwell structures. [280] Furthermore, the number of zero modes N 0 (a set of displacements of sites that result in no extension/compression of bonds) and the number of states of self-stresses N s (a set of tensions or compressions applied on bonds that results in no net forces on any sites) in Maxwell lattices can be counted by Calladine index theorem [281] : Recent studies have shown that some specific Maxwell structures have floppy modes (a zero mode that is not a trivial translation or rotation of the whole frame) at the boundaries, and these boundary modes are insensitive to local disturbances, showing similar topological protection properties as boundary states in TIs. [282] The floppy mode and states of self-stress mode can only exist in the form of boundary states that decay into the body, and also have a bulk-edge correspondence similar to that of elastic waves, that is, the boundary zero-energy mode is protected by the infinite general topological properties of energy bands. The topological phases in 1D and 2D Maxwell lattices are illustrated in Figure 30. [282] The 1D mechanical chain model ( Figure 30B) is analogy to the SSH model, in which the mass and spring correspond to lattice points A and B of the SSH model, respectively. The length of the spring is set so that the equilibrium position is θ = 0. It clearly shows that there emerges two different topological phases when the left rod tilts to left or right. The formed two domains are differentiated by their opposite topological polarizations, and the two topological phases cannot be converted to each other without closing the gap. Here, a topological polarization vector (Bravais lattice vector) characterizing the intrinsic directivity of Maxwell lattice is introduced: where a i is primitive vectors, n i is winding numbers of the phase of det Q(k) around the cycles of the BZ, and C i is a cycle of the BZ connecting k and k + b i , where b i is a primitive reciprocal vector satisfying: In one dimension, we only have to calculate one topological integral to obtain the topological polarization R T . The winding number n in the current 1D model is 0 or 1, indicating that the zero-energy mode is localized to the right or left edge of the linkage, respectively. However, in higher dimensions, we have to calculate one for each generator b i of the reciprocal lattice to obtain the topological polarization R T . For periodic trusses with units that preserve inversion symmetry, there is no nontrivial topological phase. Therefore, topological polarization cannot be generated in such a system. The corresponding topological polarization can be generated by appropriate distortion of such units and by breaking their inversion symmetry to construct different lattice vectors, so that their bulk gaps can be opened to realize the topological phase of the system. Take the deformed Kagome lattice ( Figure 30C) as an example, the topological phase of the lattice located on the left (right) side is trivial (R T = 0). To generate the topological polarization, further distortion is needed as shown in the middle of lattice. The topological polarization vector can be obtained by parameterizing the lattice: After distortion and connection, a domain wall forms wherein floppy modes and states of self-stress mode located on the left and right side of the domain wall, respectively. Similar to the topological phenomena in elastic waves, when the deformed Kagome opens the zero-frequency degeneracy except the zero point of BZ and the bulk provides a nontrivial topological phase, the zero-mode is topologically protected, as shown in Figure 30D. The topological phases can not only be accomplished by the full bulk gap but also be initiated from the Weyl points [283] of the 2D square lattice (Weyl lines [287] in 3D systems), featuring topological waves with similarity to the Weyl systems. Figure 30E and F provide an illustration of a deformed square lattice with nontrivial WPs in its spectrum. [282] Edge zero modes of the network can be transferred into the bulk at the topologically protected WPs of particular wavevectors. Another example is mechanical Weyl modes manifested in a mechanical mimic of graphene [288] when the perimeters of triangulations are limited. In addition to the regular Maxwell lattice, topological phases can also be generated in the extended Maxwell network. [284,285,289,290] A geared metamaterial in the shape of a warped Martini lattice was suggested to produce topological phases ( Figure 30G) while taking into account the elastic stability. [284] This system also supports topological Weyl modes. Topological polarization can also be formed in a modified Mikado network, in which the topological floppy modes were localized along the edge ( Figure 30H). [285] Furthermore, the topological phases can be transformed into static  [282] Copyright 2014, Springer Nature. (E) Deformed square lattices can have sinusoidal bulk zero modes (red arrows) corresponding to Weyl points (WPs), where two bands touch in the phonon dispersion (inset). Reproduced with permission. [283] Copyright 2016, American Physical Society. (F) Distorted square lattices with two WPs shown in the phonon dispersions. Reproduced under terms of the CC-BY license. [284] Copyright 2016, The Authors, American Physical Society. (G) Distorted martini lattice with nonzero polarization. (H) Modified Mikado model with a floppy mode localized on the tail of the central fiber. Reproduced with permission. [285] Copyright 2018, American Physical Society. (I) Topological polarization changed at the transitions, which results in all floppy modes being located on the top edge of the lattice. Reproduced under terms of the CC-BY license. [286] Copyright 2017, The Authors, published by Springer Nature. SSH, Su-Schrieffer-Heeger. homogeneous [286] or inhomogeneous [291] systems. For instance, in a deformed Kagome lattice with two floppy modes initially at opposite sides, the topological polarization R t can be changed by performing soft deformation, which leads to the zero modes localized at only the one side toward, as shown in Figure 30I.
In addition to Maxwell lattice, quasistatic topological phases also exist in some other metamaterials/structures, including but not limited to origami/kirigami, fishbone mechanical metamaterials, epithelial tissues, and so forth. [292][293][294][295] The topological phases of Origami and Kirigami have recently gained attention as a fresh approach to the constitution of topological metamaterials. [292] In the quasi-1D origami strip, soft mode and stiff mode can be induced on the left and right sides by carefully selecting fold pattern angles. However, to achieve topological phases in a 2D origami, the Kirigami is necessary to naturally introduce or remove constraints to generate nonzero topological polarization. Subsequently, two Kirigami heterostructures with various topological polarizations, as a result, develop topologically protected zero modes on their domain walls ( Figure 31A). Another example of topological zero edge modes can be generated in a modified fishbone mechanical metamaterial, which contributes to the realization of nonreciprocity in static systems ( Figure 31B). [294] In static mechanical systems, there exists another topological phase named topological soliton. [296][297][298] A typical illustration is the static topological soliton excited in multistable mechanical metamaterials, as shown in Figure 31C. [298] The ordered localized deformation of the metamaterial is realized under the quasistatic uniform compressive load, which is insensitive to defects. Another interesting work took into account the typical nonorientable model Möbius strips. [297] The topological floppy modes can be excited by solitons, but the number of zero modes does not satisfy the Kane-Lubensky-Maxwell-matter index ( Figure 31D).

| CONCLUSION AND OUTLOOK
Topological phononics has enabled a variety of fascinating wave-propagating phenomena that have not been realized in traditional acoustics. The combination of topology and phononics is beneficial and inspiring to each other. The acoustic system takes advantage of being feasible to fabricate, and the structure is temperature and chemical stability. The longitudinal acoustic wave lacks intrinsic spin DOF, and the classic wave takes zero-frequency limit with no confine around the Fermi level. These distinctions support the huge success of topological phononics in verifying and finding striking features that no electronic counterpart has. In acoustics, topology has contributed to various applications through robust control of boundary states. These topological F I G U R E 31 Topological phases in mechanical metamaterials. (A) Zero modes formed at the domain wall of kirigami heterostructures with different topological polarizations. The middle kirigami pattern illustrates how the quadrilateral plates, strips of triangles, and quadrilateral holes are joined by hinges. Reproduced with permission. [293] Copyright 2016, American Physical Society. (B) Nonreciprocal topological metamaterials with a fishbone structure. The angles are asymmetric (depicted in green). When applying an external load F = 0.15 N either from the left (red arrow) or right (blue arrow), the displacement always decays from right to left, indicating that the topological edge modes are localized to the right edge. Reproduced with permission. [294] Copyright 2017, Springer Nature. (C) Deformed configurations of the Block-Spring metamaterials. Reproduced under terms of the CC-BY license. [296] Copyright 2019, The Authors, published by Springer Nature. (D) Heterogeneous buckling pattern of a Möbius strip. The solid curve in the diagram shows the shape of the floppy mode bound to the solitonic excitation. Reproduced under terms of the CC-BY license. [297] Copyright 2019, The Authors, published by the American Physical Society. modes may find applications in acoustic logic devices and surface wave devices in the near future.
Topology has many extensions, and the scope of topological phononics can be further generalized to high dimensions in cooperating with synthetic space. Its achievements in the non-Hermitian, nonlinear system have remained to be explored. There is plenty of room for topological acoustic metamaterials, including but not limited to the terms listed in Figure 32.
This review mainly focuses on the boundary effect protected by a specific nontrivial gap topology. The system's topology actually manifests in other global properties of the whole band, such as fragile topology [74] and non-Abelian band topology. [299] The former is based on a definition that an insulator cannot be continuously transformed to an atomic limit without closing a gap or breaking the protecting symmetry. Kaleidoscope phononic crystals provide large freedom to design fragile topology and potentially enrich the boundary effects. The latter relies on the Euler classification of real bands instead of the Chern class. Non-Abelian topological charge emerges with non-Abelian braiding being a promising direction. In addition to fundamental studies of topology effects in phononics to develop applications or promote the understanding of the topology origin in wave dynamics, we believe its combination with other techniques and interdisciplinary fields will shine in the near future.