# Topological phononic metamaterials

Weiwei Zhu111These authors contributed equally Department of Physics, National
University of Singapore, Singapore 117542, Singapore  Weiyin Deng222These
authors contributed equally School of Physics and Optoelectronics, South China
University of Technology, Guangzhou, Guangdong 510640, China  Yang Liu333These
authors contributed equally Institute of Theoretical and Applied Physics,
School of Physical Science and Technology & Collaborative Innovation Center of
Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
Jiuyang Lu444These authors contributed equally School of Physics and
Optoelectronics, South China University of Technology, Guangzhou, Guangdong
510640, China  Zhi-Kang Lin  Institute of Theoretical and Applied Physics,
School of Physical Science and Technology & Collaborative Innovation Center of
Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
Hai-Xiao Wang  College of Physics and Technology, Guangxi Normal University,
Guilin 541004, China  Xueqin Huang555Corresponding author:
<EMAIL_ADDRESS>School of Physics and Optoelectronics, South China
University of Technology, Guangzhou, Guangdong 510640, China  Jian-Hua
Jiang666Corresponding author<EMAIL_ADDRESS>Institute of
Theoretical and Applied Physics, School of Physical Science and Technology &
Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow
University, Suzhou 215006, China  Zhengyou Liu777Corresponding author:
<EMAIL_ADDRESS>Key Laboratory of Artificial Micro- and Nano-structures of
Ministry of Education and School of Physics and Technology, Wuhan University,
Wuhan 430072, China  Institute for Advanced Studies, Wuhan University, Wuhan
430072, China

###### Abstract

The concept of topological energy bands and their manifestations have been
demonstrated in condensed matter systems as a fantastic paradigm toward
unprecedented physical phenomena and properties that are robust against
disorders. Recent years, this paradigm was extended to phononic metamaterials
(including mechanical and acoustic metamaterials), giving rise to the
discovery of remarkable phenomena that were not observed elsewhere thanks to
the extraordinary controllability and tunability of phononic metamaterials as
well as versatile measuring techniques. These phenomena include, but not
limited to, topological negative refraction, topological ‘sasers’ (i.e., the
phonon analog of lasers), higher-order topological insulating states, non-
Abelian topological phases, higher-order Weyl semimetal phases, Majorana-like
modes in Dirac vortex structures and fragile topological phases with spectral
flows. Here we review the developments in the field of topological phononic
metamaterials from both theoretical and experimental perspectives with
emphasis on the underlying physics principles. To give a broad view of
topological phononics, we also discuss the synergy with non-Hermitian effects
and cover topics including synthetic dimensions, artificial gauge fields,
Floquet topological acoustics, bulk topological transport, topological
pumping, and topological active matters as well as potential applications,
materials fabrications and measurements of topological phononic metamaterials.
Finally, we discuss the challenges, opportunities and future developments in
this intriguing field and its potential impact on physics and materials
science.

††: Rep. Prog. Phys.

###### Contents

  1. 1 Introduction
  2. 2 Fundamental aspects
    1. 2.1 Phononic metamaterials
    2. 2.2 Conventional and higher-order topology
    3. 2.3 Topological band theory
    4. 2.4 Wilson loop
    5. 2.5 Symmetry indicators
    6. 2.6 Floquet topological insulators and non-Hermitian topological insulators
    7. 2.7 Phenomenological theories: Jackiw-Rebbi theory
  3. 3 Topological phononic metamaterials
    1. 3.1 Valley and spin Hall phononic crystals
    2. 3.2 Acoustic Chern insulator and spin-Chern insulator
    3. 3.3 Weyl acoustic metamaterials
    4. 3.4 Other Weyl acoustic metamaterials
    5. 3.5 Dirac acoustic metamaterials
    6. 3.6 Nodal line and nodal surface acoustic metamaterials
    7. 3.7 Higher-order topological insulators
    8. 3.8 Non-Abelian topological states
    9. 3.9 Topological defects and bulk-defect correspondences
    10. 3.10 Acoustic topological insulators induced by projected symmetry
    11. 3.11 Mechanical metamaterials at macroscopic scales
    12. 3.12 Micron- and nano-scale mechanical metamaterials
    13. 3.13 Non-Hermitian topological phononic systems
  4. 4 Special topics
    1. 4.1 Pseudo magnetic field
    2. 4.2 Synthetic dimension
    3. 4.3 Acoustic Floquet topological insulators
    4. 4.4 Bulk transports in acoustic heterostructures
    5. 4.5 Topological pumping
    6. 4.6 Topological active matter
    7. 4.7 Acoustic skyrmions
    8. 4.8 Twisted phononic bilayers
    9. 4.9 Fractal topological acoustic metamaterials
  5. 5 Potential applications
    1. 5.1 Acoustic double-zero-index medium at Dirac-like point
    2. 5.2 Topological analog signal process for equation solving and image processing
    3. 5.3 Topological acoustic energy harvest and topological acoustic tweeters
    4. 5.4 Topological acoustic antennas
    5. 5.5 On-chip phononic topological devices.
    6. 5.6 Topological sasers.
    7. 5.7 Phononic topological logic gates
  6. 6 Methodology
    1. 6.1 Fabrication of phononic metamaterials
    2. 6.2 Measurements
  7. 7 Summary and outlooks

##  1 Introduction

Phononic metamaterials [1, 2], including acoustic metamaterials [1, 2, 3, 4,
5, 6, 7] and mechanical metamaterials [8, 9, 10], was proposed about three
decades ago and have been developed comprehensively in both fundamental
science and applications, from both theory and experiments. The concept of
phononic crystals [11, 12, 13, 14, 15], the motherboard of various phononic
metamaterials, arises when one organize phononic structures in periodic
patterns. By applying the Bloch theory on phononic waves in periodic
structures, phononic band structures can be obtained. The merit here is that
such phononic band structures are highly tunable if the phononic structures
and their periodic patterns are manipulated. This tunability gives rise to
remarkable control over acoustic waves and mechanical waves, leading to
extraordinary effects such as phononic band gaps, negative refraction,
unconventional effective parameters, versatile control of phononic waves, as
well as subwavelength imaging, trapping and waveguiding. After decades of
study, phononic metamaterials became a material platform with fantastic
controllability and reliable measurement techniques.

In the past years, topological phononic metamaterials have attracted
tremendous research interest [16, 17, 18, 19, 20]. The field has been growing
rapidly ever since it is born. Studies on topological phononic metamaterials
aim to exploit topological physics—an important branch of condensed matter
physics where the topological properties of energy bands play central roles
[21, 22, 23, 24, 25, 26, 27, 28, 29]—to explore new phenomena, materials, and
their applications. The development of topological phononic metamaterials have
brought many unprecedented effects into phononics and acoustics. For instance,
the Zak phases and topological reflection, Weyl phonons and topological
negative refraction, topological ‘sasers’ (i.e., the phonon analog of lasers),
non-Hermitian topological phononic metamaterials, topological Wannier cycles,
bulk-defect correspondences, higher-order topological phases of phonons, non-
Abelian topological states of phonons, phononic Dirac cones and double zero
effective medium, to list but only a few. The field is still growing rapidly
with lots of new ideas emerging in both fundamental and application aspects.
The purpose of this review is to introduce the basic notions, models,
pictures, and methods as well as the main progresses, opportunities, and
challenges in this field so that readers from a broad range of disciplines can
understand the main pages of this field. In this way, interdisciplinary
studies and new ideas can continuously fuel researches in this direction.

The motivations of harnessing band topology in phononic metamaterials came
from two main perspectives: First, phononic metamaterials are particularly
appealing for the realization and observation of various topological
phenomena. Thanks to their excellent tunability, topological phononic
metamaterials can be designed into almost any geometry to realize various
topological phases, ranging from valley- and spin-Hall phases to Weyl, Dirac,
nodal line, higher-order, and fragile topological phases. Furthermore,
mechanical systems provide extraordinary ability of detecting the bulk and
boundary phononic states (both the wavefunctions and the spectra) with
excellent position, wavevector, and frequency resolutions. Second, the
developments of topological phononic metamaterials greatly enrich the study of
metamaterials for unprecedented control of phononic waves and mechanical
energies. These two motivations have been the main driving power for the
development of topological phononic metamaterials, leading to researches
ranging from the exploration of extreme topological phases that are hard to
realize in condensed matter systems, to new ideas and phenomenon that have not
yet been proposed in any other field, to cutting edge applications such as
high-performance mechanical sensors, on-chip information processing based on
integrated acousto-electronic systems, high-quality resonators, and robust
frequency combs, etc. With these remarkable progresses and enormous research
interest, the study on topological phononic metamaterials has been moving
rapidly for years.

This review article is structured as follow: In Sec. II. we introduce the
basic aspects of phononic metamaterials, the fundamental concepts and methods
for topological phononics as well as some phenomenological theories such as
the Jackiw-Rebbi theory. In Sec. III, we elaborate on various topological
phononic metamaterials, forming four main categories: First, topological
acoustic metamaterials including acoustic valley Hall, spin Hall, Chern
insulators, and their generalizations, higher-order topological phases,
topological semimetals, non-Abelian topological states, topological defects,
and topological states protected by projective symmetries. Second, mechanical
metamaterials at macroscopic scales such as spring mass systems. Third,
micron-and nano-scale mechanical metamaterials that are appealing for on-chip
applications. Fourth, non-Hermitian topological phononic systems that exhibit
unconventional properties. In addition, we also discuss specific topics such
as synthetic dimensions, artificial gauge fields, Floquet topological phononic
systems, and bulk topological transport. Other topics covered here include
Thouless pumping of phonons, topological active matters, acoustic skyrmions
etc. For completeness, in Sec. IV, we summarize the commonly used fabrication
and measurement methods for the study of topological phononic metamaterials.
In Sec. V, we discuss the potential applications of topological phononic
metamaterials. Finally, we give outlooks with discussions on future challenges
and possible developments in Sec. VI. Alongside with previous reviews [16, 17,
18, 19, 20], this article intended to provide a systematic and the most up-to-
date review on topological phononic metamaterials.

##  2 Fundamental aspects

###  2.1 Phononic metamaterials

Topological phononic metamaterials are periodically distributed arrays of
artificial phononic atoms, which usually support robust topological edge
states [16, 17, 18, 19, 20]. They are analogies of topological states for
electrons in periodic potential. It is well known that the behavior of
electrons scattered by periodic potential can be understood by different
physical picture like tight-binding picture, free electron gas picture, etc
[30]. As the analogies, the topological phononic metamaterials can be
classified into three classes corresponding to different physical pictures,
based on the types of artificial atoms (resonators or scatterers) and the
couplings style (near field couplings or far field couplings), as shown in
Fig. LABEL:classification. Figure LABEL:classification(a) shows the first
class of the phononic metamaterials, where the waves are confined in the
coupled phononic cavities [31]. The confined waves in different cavities are
coupled by near field through coupling tubes with small radius. The phononic
waves in such system correspond to electrons in periodic deep well shown in
Fig. LABEL:classification(d) and can be described by the tight-binding
picture. In such picture, the waves were considered to be localized in the
deep well. The electrons can hop between different wells by quantum tunneling
and form pass bands. Figure LABEL:classification(b) is the one-dimensional
phononic crystal which is composed of AB layers with different cross-sectional
areas [32]. One plane wave will be scattered when it propagates from one layer
to another layer. The phononic waves in such system correspond to electrons in
periodic shallow well or electrons of high energy shown in Fig.
LABEL:classification(e) known as free electron gas picture. In such picture,
the waves were considered to propagate and be scattered at the edges of
potential wells. The destructive interference between forward and scattering
waves forms the forbidden bands. Figure LABEL:classification(c) is the locally
resonant phononic metamaterial [1, 33]. The side branch tubes are local
resonators. The local resonators can be excited by the propagating waves in
the main waveguide and then leak waves into the main waveguide. Such system
corresponds to a light-matter interaction system shown in Fig.
LABEL:classification(f), where the electrons are bounded by positively charged
ions and interact with each other by radiating light. In such picture, the
electrons in ground state can absorb photons from waveguide to excited states
and the excited electrons can radiate photons into the waveguide. The
interaction between forward propagating photons and radiating photons forms a
band structure for quasi-particles as the hybridization of electrons and
photons.

The three classes of topological phononic metamaterials have their own
advantages. The first class of coupled phononic cavities provides a nice
platform to realize topological states described by the tight-binding models.
The couplings between cavities are realized by connecting tubes. Both the
nearest-neighbor couplings and long-range couplings can be easily obtained.
The coupling strength is tuned by changing the radius of the connecting tubes.
Even negative couplings can be realized by connecting two local positions of
two cavities with opposite field phase [34, 35, 36, 37]. Fig.
LABEL:classification(g) shows the positive coupling between two cavities with
$p_{z}$ orbitals and Fig. LABEL:classification(h) shows the negative coupling.
Fig. LABEL:classification(i) shows the negative coupling between one cavity
with $p_{x}$ orbital and the other cavity with $p_{y}$ orbital. Besides, the
negative coupling also can be realized by inceasing the length of coupling
tubes [38]. The phononic crystal provides a nice platform to realize
topological states in scattering (continuous) system. The topological
properties of the system are determined by the symmetry of lattice, the
symmetry of individual scatterer and the relative position between different
scatterers. Fruitful topological phases have been obtained by only rotating
the scatterers [39, 40]. Its feasibility and tunability have been widely
proven both also in two-dimensional systems and three-dimensional systems. The
locally resonant phononic metamaterial provides a nice platform to realize
subwavelength topological states whose sizes are more compact. Besides, due to
the couplings between cavities are far field, it also provides a nice platform
to study topological states in open system. The topological properties of the
system can be tuned by changing resonant frequency of the cavities or the
relative position between the cavities.

###  2.2 Conventional and higher-order topology

To introduce conventional topological insulators and higher-order topological
insulators, we first introduce the concept of codimension, which is the
difference between the dimensions of the system and the topological boundary
states. The codimension of conventional topological insulators is one, i.e.,
$n$-dimensional system supports $n-1$-dimensional gapless boundary states. The
Chern insulator [41, 42], $Z_{2}$ topological insulator [43], valley
topological insulator [44, 45, 46, 47, 48], etc. belong to conventional
topological insulators. The topological properties of conventional topological
insulators are protected by bulk topology. Higher-order topological insulators
are those states with codimension larger than one [49, 50,
phys.rev.b.97.205136khalaf, 52, 53, 54, 55, 56, 57, 58]. The two-dimensional
higher-order topological insulator has gapped edge states but supports
topological corners states. $n$-dimensional system supports $n-m$-dimensional
($m>1$) boundary states. The higher-order topological insulators can be
classified into intrinsic and extrinsic ones [59]. The intrinsic higher-order
topological insulator is protected by bulk topology. Spatial symmetry plays an
important role in it. The extrinsic higher-order topological insulator is
protected by boundary states. The higher-order topology can be more general.
Gapless states can support topological corner states in the bulk continuum
[60]. The higher-order Dirac/Weyl semimetals are also proposed as the higher-
order topologies [61, 62].

###  2.3 Topological band theory

We have roughly introduced three classes of topological phononic metamaterials
in previous part. Although they are corresponding to different physical
pictures, all of them have periodic structures described by the periodic
Hamiltonian

$\hat{H}(\mathbf{r})=\hat{H}(\mathbf{r}+\mathbf{R}),$ (1)

where $\mathbf{R}$ is the Bravais lattice vector. For the coupled phononic
cavities, $\hat{H}(\mathbf{r})$ usually can be described by a sparse matrix,
$H_{mn}=\langle m|\hat{H}(\mathbf{r})|n\rangle$. $H_{mn}$ is the near field
couplings between the $m^{\text{th}}$ cavity and the $n^{\text{th}}$ cavity.
For the phononic crystals, $\hat{H}(\mathbf{r})$ are usually described by
partial differential operators [63], e.g. $-\rho^{-1}\kappa\nabla\cdot\nabla$,
where $\nabla$ is the Nabla operator, $\rho$ is the mass density and $\kappa$
is the bulk modulus. For the locally resonant phononic metamaterials,
$\hat{H}(\mathbf{r})$ usually can be described by a dense matrix [64],
$H_{mn}=\langle m|\hat{H}(\mathbf{r})|n\rangle$. $H_{mn}$ is the far field
couplings between the $m^{\text{th}}$ cavity and the $n^{\text{th}}$ cavity.
The spectrum of the system can be obtained by solving the following eigen-
equation,

$\hat{H}(\mathbf{r})\psi(\mathbf{r})=E\psi(\mathbf{r}).$ (2)

Due to $\hat{H}(\mathbf{r})$ is periodic in real space, the crystal momentum
is a good quantum number. According to the Bloch theory, the eigenstates of
the system satisfy the following expression,

$\psi_{n\mathbf{k}}(\mathbf{r})=u_{n\mathbf{k}}(\mathbf{r})e^{i\mathbf{k\cdot
r}},$ (3)

where $u_{n\mathbf{k}}(\mathbf{r})$ is the periodic part of the Bloch wave
function. The energy bands of the system can be obtained from the following
equation,

$H(\mathbf{k})u_{n}(\mathbf{k})=E_{n}(\mathbf{k})u_{n}(\mathbf{k}).$ (4)

Here
$H(\mathbf{k})=e^{-i\mathbf{k}\cdot\mathbf{r}}\hat{H}(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}$
is obtained from the Fourier transformation of the real space Hamiltonian.
$H(\mathbf{k})$ or equivalently its eigenvalues $E_{n}(\mathbf{k})$ and
eigenvectors $u_{n}(\mathbf{k})$ defines the band structure.

$E_{n}(\mathbf{k})$ also known as energy bands determine the most important
transport properties of waves in the bulk, e.g., the waves are extended in the
pass band and localized in the forbidden band. Recent studies on topological
states pay more attention to the eigenvectors $u_{n}(\mathbf{k})$. Two systems
with same energy bands $E_{n}(\mathbf{k})$ can have dramatically different
topological properties, which are distinguished by various topological
invariants determined by the global properties of the Bloch waves. For
example, the usual Berry phase [65] of the $n^{\text{th}}$ band can be
obtained from the periodic part of the Bloch wave function by the definition

$\theta_{n}=\oint_{\mathcal{C}}d\mathbf{k}\cdot\langle
u_{n}(\mathbf{k})|i\bigtriangledown_{\mathbf{k}}|u_{n}(\mathbf{k})\rangle,$
(5)

where $\mathcal{A}_{n}(\mathbf{k})=\langle
u_{n}(\mathbf{k})|i\bigtriangledown_{\mathbf{k}}|u_{n}(\mathbf{k})\rangle$ is
the Berry connection corresponding to a vector potential in momentum space and
$\mathcal{C}$ is a closed path in momentum space. The
$\mathcal{A}_{n}(\mathbf{k})$ is a gauge dependent quantity. We can see that
the eigenvector has an arbitrary phase choice
$|u^{\prime}_{n}(\mathbf{k})\rangle=e^{i\phi_{n}(\mathbf{k})}|u_{n}(\mathbf{k})\rangle$
known as gauge transformation. After the transformation, the corresponding
Berry connection is changed to $\mathcal{A}^{\prime}_{n}(\mathbf{k})=\langle
u^{\prime}_{n}(\mathbf{k})|i\bigtriangledown_{\mathbf{k}}|u^{\prime}_{n}(\mathbf{k})\rangle=\mathcal{A}_{n}(\mathbf{k})-\nabla_{\mathbf{k}}\phi_{n}(\mathbf{k})$.
However the Berry phase $\theta_{n}$ is gauge independent and has physical
meaning. For one-dimensional model, $\mathcal{C}$ is the first Brillouin zone
and $\theta_{n}$ is the Zak phase of the $n^{\text{th}}$ band. The Zak phase
determines the Wannier centers of the system and corresponds to the
polarization. Specially, when the system supports inversion symmetry, the Zak
phase is quantized and only take values $0$ or $\pi$ [21].

From the Berry connection, we can define the Berry curvature
$\mathbf{\Omega}_{n}(\mathbf{k})=\nabla\times\mathcal{A}_{n}(\mathbf{k})$. The
Berry curvature is gauge independent and corresponds to magnetic field in
momentum space. The well-known Chern number for two-dimensional system can be
obtained by the integration of the Berry curvature in the whole first
Brillouin zone,

$2\pi
C_{n}=\int_{\mathrm{BZ}}d^{2}\mathbf{k}\cdot\mathbf{\Omega}_{n}(\mathbf{k}).$
(6)

The Chern number can be connected with the Berry phase by the following
relation [66],

$\begin{split}2\pi
C_{n}=&-\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}dk_{x}dk_{y}(\partial_{k_{x}}\mathcal{A}_{n}^{y}-\partial_{k_{y}}\mathcal{A}_{n}^{x})\\\
=&\int_{-\pi}^{\pi}dk_{y}\partial_{k_{y}}\left(\int_{-\pi}^{\pi}dk_{x}\mathcal{A}_{n}^{x}\right)\\\
=&\int_{-\pi}^{\pi}d\theta_{n}(k_{y}).\end{split}$ (7)

Here $\theta_{n}(k_{y})=\int_{-\pi}^{\pi}dk_{x}\mathcal{A}_{n}^{x}$ is the
Berry phase calculated from the one-dimensional integration of
$\mathcal{A}_{n}^{x}(k_{x},k_{y})$ along the $k_{x}$ direction for each fixed
$k_{y}$. We notice the Chern number is just the winding of the Berry phase
$\theta_{n}(k_{y})$ as a function of $k_{y}$. Such expression has more
physical meaning. We know that the Berry phase $\theta_{n}(k_{y})$ determines
the Wannier center of one-dimensional system and also determines the
topological edge states. $\theta_{n}(k_{y})$ is also called the Wannier bands.
The nonzero winding of the Berry phase means the gapless topological edge
states. According to the bulk-boundary correspondence, the Chern number
determines the number of topological chiral edge states. One important
property of topological edge states is that they are robust against disorders.
Such property can be used to design robust topological devices.

Although the definitions of the Berry phase in Eq. 5 and the Chern number in
Eq. 7 are clear, they are still hard to be calculated practically. The reason
is that the derivative of $|u_{n}(\mathbf{k})\rangle$ to $\mathbf{k}$ requires
$|u_{n}(\mathbf{k})\rangle$ to be continuous with $\mathbf{k}$, which is
usually hard to realize in calculation. Systems with multiple bands increase
the calculation complexity further. Besides, the Berry connection is a gauge
dependent quantity, which also brings difficulties in numerical calculation.
Latter, we will introduce the Wilson loop approach, which is compatible with
numerical implementation to calculate the topological invariants. Symmetry
indicators are another way to distinguish topological states. Before that, we
first introduce a concrete model to show the difference between topological
trivial states and nontrivial states. Latter we will also use this model to
show the power of the Wilson loop approach and symmetry indicators. The model
is the breathing honeycomb lattice shown in Fig. LABEL:dimerisedhoneycomb(a)
and is described by the Hamiltonian [67]

$H=\sum\limits_{\langle i,j\rangle}c_{ij}a_{i}^{\dagger}a_{j},$ (8)

where $a_{i}^{\dagger}$ ($a_{i}$) is the creation (annihilation) operator for
a particle at the $i$th site, and $c_{ij}$ is the nearest neighbor coupling.
Each unit cell contains six sites. The couplings in one unit cell are
$c_{ij}=c_{\mathrm{int}}$ and the couplings between unit cells are
$c_{ij}=c_{\mathrm{ext}}$. The model only contains the real nearest-neighbor
coupling, which is friendly to experiment in phononics. As we show below, such
simple model can describe a first-order topological insulator or a second-
order topological insulator, depending on the parameters. The topological
properties of the model can be understood by Wilson loop or symmetry
indicators, so that it can be treated as a standard example to introduce basic
concepts for topological phononics.

By doing a Fourier transformation to real-space Hamiltonian in Eq. 8, we
obtain the Hamiltonian in momentum space as

$H(\mathbf{k})=\left(\begin{array}[]{cccccc}0&c_{\mathrm{int}}&0&c_{\mathrm{ext}}e^{i\mathbf{k}\cdot\mathbf{a}_{1}}&0&c_{\mathrm{int}}\\\
c_{\mathrm{int}}&0&c_{\mathrm{int}}&0&c_{\mathrm{ext}}e^{i\mathbf{k}\cdot\mathbf{a}_{2}}&0\\\
0&c_{\mathrm{int}}&0&c_{\mathrm{int}}&0&c_{\mathrm{ext}}e^{i\mathbf{k}\cdot\mathbf{a}_{3}}\\\
c_{\mathrm{ext}}e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}&0&c_{\mathrm{int}}&0&c_{\mathrm{int}}&0\\\
0&c_{\mathrm{ext}}e^{-i\mathbf{k}\cdot\mathbf{a}_{2}}&0&c_{\mathrm{int}}&0&c_{\mathrm{int}}\\\
c_{\mathrm{int}}&0&c_{\mathrm{ext}}e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}&0&c_{\mathrm{int}}&0\end{array}\right),$
(9)

where $\mathbf{k}=(k_{x},k_{y})$ is the Bloch momentum,
$\mathbf{a}_{1}=(0,a)$, $\mathbf{a}_{2}=(\sqrt{3}a/2,a/2)$,
$\mathbf{a}_{3}=(\sqrt{3}a/2,-a/2)$, and $a$ is the lattice constant. The
spectrum of the system can be obtained by solving the eigen equation,

$H(k)u_{n}(\mathbf{k})=E_{n}(\mathbf{k})u_{n}(\mathbf{k}),$ (10)

where $u_{n}(\mathbf{k})$ is the periodic part of the Bloch wave function, and
$n$ is the index of the band.

Two configurations are studied, one is the topological trivial case with
$c_{\mathrm{int}}>c_{\mathrm{ext}}$ shown in Fig. LABEL:dimerisedhoneycomb(a),
and the other is the topological nontrivial case with
$c_{\mathrm{int}}<c_{\mathrm{ext}}$ shown in Fig. LABEL:dimerisedhoneycomb(b).
The spectrum of the strip structure with periodic boundary condition along the
$x$ direction and open boundary condition along the $y$ direction is studied.
For the topological trivial case, there is no states in the band gap as shown
in Fig. LABEL:dimerisedhoneycomb(c), while for the topological nontrivial
case, there are topological edge states in band gap as shown in Fig.
LABEL:dimerisedhoneycomb(d). Note that each edge supports a pair of counter-
propagating edge states. Besides, the system is a weak topological insulator
in the sense that the topological edge states are gapless only in the zigzag
edge but become gapped for other edge boundaries.

###  2.4 Wilson loop

Wilson loop approach is a nice tool to calculate the topological invariant,
which is gauge invariant and compatible with numerical implementations [66].
The method is quite general and can be used to describe both tight-binding
models and continuous models [68]. It is also convenient to use it to describe
multi-band system like introduced in Fig. LABEL:dimerisedhoneycomb. Here we
first provide the concept of the Wilson loop and then apply it to describe the
topological properties of the breathing honeycomb lattice discussed above.
Considering a matrix $M^{k_{i},k_{i+1}}$, whose elements are obtained from

$M_{nn^{\prime}}^{k_{i},k_{i+1}}=\langle
u_{n,k_{i}}|u_{n^{\prime},k_{i+1}}\rangle,$ (11)

where $n$ and $n^{\prime}$ are band indices covering all the $N$ bands below
the concerned band gap, and $k_{i}$ and $k_{i+1}$ are two closer points on a
loop in Brillouin zone. The Wilson loop can be obtained from the matrix
product of a class of $M^{k_{i},k_{i+1}}$,

$W=\prod_{i=1}^{N}M^{k_{i},k_{i+1}},$ (12)

where $k_{1},k_{2},\ldots,k_{N}$ form a closed loop in the Brillouin zone and
$k_{i+N}=k_{i}$. For one-dimensional problem, the Berry phases of each band
are given by

$\theta_{n}\equiv i\log{w_{n}},$ (13)

where $w_{n}$ is the $n^{\text{th}}$ eigenvalue of $W$. For higher-dimensional
system, we can consider the Berry phase along one direction $k_{\parallel}$ as
a function of $k_{\perp}$, i.e., $\theta_{n}(k_{\perp})$. For example, the
well-knownn Chern number in two-dimensional system can be obtained by the
winding of $\theta_{n}(k_{y})$ as introduced in Eq. 7.

Next we apply the Wilson loop approach to study the topological properties of
the breathing honeycomb lattice. The first Brillouin zone of the breathing
honeycomb lattice is a regular hexagon shown in Fig. LABEL:wilson(a).
$\Gamma$, $\mathrm{M}$, $\mathrm{K}$ and $\mathrm{K}^{\prime}$ are the high
symmetric momentum points. From Fig. LABEL:dimerisedhoneycomb, we notice that
the unit cell is enlarged when considering the strip structure along the
zigzag direction. The enlarged unit cell corresponds to a reduced first
Brillouin zone which is shown in Fig. LABEL:wilson(a) by blue color. To
describe its topological properties, we need to consider the bulk bands of the
enlarged unit cell. The original unit cell contains six sites so that there
are six bulk bands. When considering the enlarged unit cell, the bulk bands
are fold and there are totally twelve bulk bands with six below the zero gap
and six up to the zero gap as shown in Fig. LABEL:wilson(b). The Berry phases
for the lower six bands are calculated by the Wilson loop approach. The
results for the topological trivial model and topological nontrivial model are
shown in Figs. LABEL:wilson(c) and LABEL:wilson(d), respectively. We notice
for the topological trivial model, the Berry phases for all six bands are
located around 0 or $2\pi$, which can be continuously transformed to 0 or
$2\pi$, thus is just an atomic insulator. However, for the topological
nontrivial model, the Berry phases of the two bands are crossing around $\pi$,
which is corresponding to a nonzero polarization and is the topological origin
of the topological edge states in Fig. LABEL:dimerisedhoneycomb(d). The Berry
phase crossing and the topological edge states here have similar behavior to
the topological insulators with time-reversal symmetry. The topological edge
states in Fig. LABEL:dimerisedhoneycomb(d) are double degenerate. However,
they are different. Here the Wannier bands in Fig. LABEL:wilson(d) are still
gapped while the Wannier bands of topological insulator are gapless. The
reason is that here the topological breathing honeycomb lattice is a weak
topological insulator and the topological edge states only exist along
specific directions. In addition, the topological edge states in Fig.
LABEL:dimerisedhoneycomb(d) do not connect with the bulk states.

###  2.5 Symmetry indicators

Above we have shown that the topological states can be described by the
Wannier bands calculated from the Wilson loop approach. However, the
calculation is still inconvenient. Fu and Kane have shown one simple way to
calculate the topological invariant of topological insulators with inversion
symmetry, which only consider the inversion symmetry eigenvalues at high
symmetric momentum points [69]. This method has inspired the development of
symmetry indicators of band topology, which can be used to deal with various
symmetries and different topological states [70, 71]. The development of
symmetry indicators enables the diagnosis of topological materials [72, 73]
and further inspires new concepts, such as higher-order topological insulators
and fragile topological insulators [74].

Atomic insulators with crystalline symmetry have symmetric and exponentially
localized Wannier functions. Given a crystal symmetry, the band representation
for all possible atomic insulators at high symmetric momentum points can be
obtained by placing different orbitals at different maximal Wyckoff positions.
Insulators with specific symmetry can be classified into different topological
phases by comparing their symmetry eigenvalues at high symmetric momentum
points with atomic insulators. The insulators, which can be treated as sum of
atomic insulators, are classified into normal insulators. Those can be
smoothly deformed into a linear superposition of atomic insulator with some
coefficients being negative are classified into fragile topological
insulators. The conventional topological insulators are more stable, and they
can not be transformed into atomic insulators without breaking symmetry or
closing band gap, so that can not be expressed as a linear superposition of
atomic insulators. Due to the existence of symmetry, the normal insulator can
be further subdivided into different topological phases. For example, the
atomic insulators with Wannier function at boundary of unit cell are
topologically different from the atomic insulators with Wannier functions at
the center of unit cell. In this sense, we also can treat them as topological
nontrivial states although they usually do not support gapless topological
edge states. Those topological phases can be roughly distinguished by checking
the symmetry eigenvalues at high symmetric momentum points [70, 71].

We still use the breathing honeycomb lattice shown in Fig.
LABEL:dimerisedhoneycomb as an example. The Hamiltonian in Eq. 9 supports six
fold rotation symmetry

$C_{6}H(\mathbf{k})C_{6}^{{\dagger}}=H(R_{6}\mathbf{k}),$ (14)

where $C_{6}$ is a six by six matrix with elements $C_{m,\mod(m+1,6)}=1$ and
$R_{6}$ is a matrix that rotates the crystal momentum by $2\pi/6$ as

$R_{6}=\left(\begin{array}[]{cc}\cos(2\pi/6)&\sin(2\pi/6)\\\
-\sin(2\pi/6)&\cos(2\pi/6)\end{array}\right).$ (15)

The maximal Wyckoff positions for the breathing honeycomb lattice are shown in
Fig. LABEL:symmetry(a). There are one orbital at the center of the unit cell
$1a$ whose little group is $C_{6}$, two orbitals at position $2b$ whose little
group is $C_{3}$, and three orbitals at position $3c$ whose little group is
$C_{2}$. In momentum space, the first Brillouin zone is shown in Fig.
LABEL:symmetry(b). $\Gamma$, M and K are three high symmetric momentum points.
The Bloch Hamiltonian supports $C_{6}$ symmetry at $\Gamma$, $C_{2}$ symmetry
at M and $C_{3}$ symmetry at K. We consider the lowest three bands so there
are three orbitals per unit cell. There are three possible atomic insulators
which can be equivalent to it: the first one is built by placing three
orbitals at position $3c$, the second by placing one orbital at $1a$ and two
orbitals at $2b$, and the third by placing three orbitals at $1a$. The $C_{2}$
symmetry eigenvalues for $c_{i}>c_{o}$ are $-1,+1,+1$ at $\Gamma$ point and
$-1,+1,+1$ at M point. It is equivalent to one $p$ orbital and two $s$
orbitals at $1a$. So the charge is distributed at the center of the unit cell,
as shown in Fig. LABEL:symmetry(c). Such phase is topologically trivial. The
$C_{2}$ symmetry eigenvalues for $c_{i}<c_{o}$ are $-1,-1,-1$ at $\Gamma$
point and $-1,+1,+1$ at M point. It is equivalent to three $p$ orbitals at
$3c$. The charge distribution is shown in Fig. LABEL:symmetry(d). We notice
there are charges on the edges and corners. They contribute to edge states and
corner states. So the breathing honeycomb lattice with $c_{i}<c_{o}$ is still
an atomic insulator, but it is topologically nontrivial with $c_{i}>c_{o}$.

###  2.6 Floquet topological insulators and non-Hermitian topological
insulators

Topological states have been extended to periodically driven system, which are
known as Floquet topological insulators [86, 76, 77, 78, 79, 80, 81, 82, 83,
84, 85]. Different from the static system, the Hamiltonian of a Floquet system
is time dependent and periodic in time $H(t)=H(t+T)$, where $T$ is the driving
period. Although the energy is not a conserved quantity any more, the
quasienergy is conserved. We seek quasistationary solutions that have the form
$|\psi(\mathbf{k},t)\rangle=e^{-i\varepsilon(\mathbf{k})t}|\phi(\mathbf{k},t)\rangle$,
where $|\phi(\mathbf{k},t)\rangle$ is periodic in time with period $T$. The
quasienergy band $\varepsilon(\mathbf{k})$ can be obtained from the
eigenequation,

$U_{T}(\mathbf{k})|\phi(\mathbf{k},t)\rangle=e^{-i\varepsilon(\mathbf{k})T}|\phi(\mathbf{k},t)\rangle,$
(16)

where
$U_{T}(\mathbf{k})\equiv\mathcal{T}\exp[-i\int_{0}^{T}H(\mathbf{k},\tau)d\tau]$
is the Floquet operator, and $\mathcal{T}$ denotes the time-ordering operator.

At the beginning, Floquet engineering was used as a method to tune the
topological properties of band structures. A normal insulator can be driven
into topological nontrivial states by Floquet engineering [75]. Most Floquet
topological states can be described by the topological invariants as the
static case. The Floquet system can be treated as a static system with
effective Hamiltonian $H_{\text{eff}}(\mathbf{k})\equiv\frac{i}{T}\log
U_{T}(\mathbf{k})$. Later studies show that Floquet system can support
topological states without static counterpart. From Eq. 16, we notice
$U_{T}(\mathbf{k})$ is an unitary matrix so that $\varepsilon(\mathbf{k})$ is
periodic in frequency domain with period $2\pi/T$. It means the Floquet system
can support fully gapless states [79]. The unitary $U_{T}(\mathbf{k})$ is
topologically nontrivial if it can not be continuously transformed into an
identity matrix. Besides, Floquet system also supports topological states that
are described by nontrivial topology of time evolution operator
$U(\mathbf{k},t)\equiv\mathcal{T}\exp[-i\int_{0}^{t}H(\mathbf{k},\tau)d\tau]$.
The most famous example is the anomalous Floquet topological insulator, which
supports chiral edge states with zero Chern number [77]. Anomalous Floquet
higher-order topological insulator is another example, which supports
topological corner states with vanishing dipole and quadrupole polarizations
[87, 88, 89, 90, 91, 92]. We notice the Floquet operator $U_{T}(\mathbf{k})$
is unitary matrix which is similar to the scattering matrix. Their eigen-
equations are equivalent to each other. So the Floquet topological states can
be mimicked by scattering systems [93, 94, 95, 96, 97, 98]. Phononic coupled
ring resonator system is a nice platform to simulate Floquet topological
insulator, where the scattering process plays the role of time. Besides, the
coupled waveguide system can also be used to mimic Floquet phenomena, where
the propagation along specific direction in space plays the role of time [99,
100, 101, 102, 103, 104, 105].

Non-Hermitian topology is another direction that attracts a lot of attention
recently. Non-Hermitian band theory is widely different from the Hermitian one
[106, 107, 108, 109]. First, the bulk spectrum of non-Hermitian system highly
depends on the boundary condition. It breaks the usual bulk-boundary
correspondence [110, 111]. Second, the bulk spectrum of non-Hermitian system
is typically complex so that different kinds of band gaps can be defined: a
non-Hermitian Hamiltonian is defined to have a point gap if and only if its
complex-energy bands do not cross a reference point in the complex-energy
plane; and a non-Hermitian Hamiltonian is defined to have a line gap if and
only if its complex-energy bands do not cross a reference line in the complex-
energy plane [112, 113]. While the line gap topology can be transformed to a
Hermitian counterpart without closing the band gap, the point gap is unique to
non-Hermitian system. A topological nontrivial point gap with nonzero winding
number contributes to the non-Hermitian skin effect [114, 115], causing all
the bulk states localized under open boundary condition [110, 116, 117, 118,
119, 120, 121, 111, 122, 123, 124]. While the non-Hermitian skin effect itself
has topological origin from point gap, it also affects the usual band topology
for line gap. The usual bulk-boundary correspondence does not work any more
for non-Hermitian system with non-Hermitian skin effect. Non-Bloch topological
band theory has been developed to recover the bulk-boundary correspondence for
non-Hermitian system [125, 115, 126, 127, 128, 458, 130, 131, 132, 133, 134,
135, 136, 137, 138]. Except the non-Hermitian skin effect, other non-Hermitian
topological states like non-Hermitian Chern bands [125, 139, 140, 141, 142],
hybrid skin-topological modes [143, 144, 145, 146, 147, 148], critical non-
Hermitian skin effect [149] and second-order non-Hermitian skin effect [150,
151, 152, 153] have been widely studied.

###  2.7 Phenomenological theories: Jackiw-Rebbi theory

Dirac equation proposed to behave the relativistic and quantum mechanical
electrons, now is mainly applied in condensed matter physics where it is
widely extended to describe the low-energy excitations. The solution of Dirac
equation induces the problem of negative energy modes, because physical
electrons only carry positive energy. Based on this contradiction, Dirac
speculated the existence of anti-electrons which was demonstrated by the
experimental discovery of positrons in 1932 [154]. The original Dirac equation
is written as follows,

$(\bm{\alpha}\cdot\bm{p}+\beta{m})\Psi=i\frac{\partial\Psi}{\partial t}.$ (17)

Here, coefficients $\bm{\alpha}$ and $\beta$ are matrices, which is a
significant feature of the Dirac equation. $\bm{p}$ is the momentum operator
with $\bm{p}=i\nabla$, and $m$ represents the mass term. The mass term of the
normal Dirac equation is homogenous and position-independent. However, the
defects or disturbances in the system change the constant mass term. These
inhomogenous mass terms arising from defects or disturbances can seem like an
effective phonon field implemented on the system.

In 1976, Jackiw and Rebbi obtained a soliton state with zero energy by solving
the Dirac equation based on a soliton-monopole system when the Fermi field is
present [155, 156]. This means that the inhomogenous and position-dependent
mass term can give rise to some intriguing phenomena, as the soliton states
emerge on the domain wall between two distinct lattices with opposite mass
terms. These soliton states as a significant feature of Jackiw-Rebbi model can
be realized in acoustic systems through proper mapping.

Several recent papers reveal the method of realizing the acoustic analogues of
Jackiw-Rebbi model. For instance, Xiao et al. design a sandwich-like
cylindrical waveguide structure [32]: each acoustic unit cell has two wider
tubes (tube-A) with a narrower tube (tube-B) inserted in (Fig.
LABEL:JRdminone-dimensional(a)). They demand that the longitudinal length of
each unit cell is a constant 8.5cm, which means the adjustable parameter is
only the length difference $\triangle d=(d_{\text{A}}-d_{\text{B}})/2$ between
tube-A and tube-B. By redistributing the lengths of tube-A and tube-B, they
realize a band inversion of the second gap at the critical point $\triangle
d$=0.49cm, where unit cells at its both sides have opposite effective mass
terms. A $\mathbb{Z}_{2}$ symmetric geometry phase called as Zak phase
[21]—the integral of Berry connection, is an excellent tool to distinguish
topological properties between unit cells with opposite mass terms. As shown
in Fig. LABEL:JRdminone-dimensional(a), they combine two topologically
different unit cells (S1 and S2) with opposite mass terms together to obtain a
domain wall structure. As the statement of Jackiw-Rebbi theory, there is a
soliton state localized on the domain wall. Such a soliton state is
successfully observed by Xiao et al. in the experiment and its acoustic
response is shown in Fig. LABEL:JRdminone-dimensional(b). In another work, Li
et al. propose a one-dimensional periodic acoustic system in exact analogy
with the Su-Schrieffer-Heeger model [157, 158], which formerly depicts two
stable arrays of carbon atoms in polyethylene. The unit cell of the acoustic
system is composed of two resonators and two junction tubes connecting them.
The effective opposite mass term can be obtained through exchanging the radii
of two junction tubes which control the intercell and intracell hoppings. A
domain wall constructed through unit cells A and B with opposite mass terms is
shown in Fig. LABEL:JRdminone-dimensional(c). Similarly, there is an in-gap
soliton state localized on the domain wall (Fig. LABEL:JRdminone-
dimensional(d)). This soliton state is confirmed by Li et al. through
transmission scanning and they verify the robustness of this soliton state.
For mechanical systems, Vitelli et al. also design a similar structure
composed of a periodic arrangement of alternating massless rigid rotors,
constrained to rotate about fixed pivot points, around an equilibrium angle
$\bar{\theta}$ at odd-numbered sites and $\pi-\bar{\theta}$ at even-numbered
sites [159]. On the domain wall between rotors in right-leaning and left-
leaning states (corresponds to distinct winding number in Ref. [160]), a
soliton state appears as expected. Both above works focus on one-dimensional
domain wall systems and acquire the similar result that, a soliton state
emerges on the domain wall predicted by Jakiw-Rebbi theory. And that prompts
us to consider, what is the presentation of Jackiw-Rebbi model in a two-
dimensional system.

In one-dimensional systems, the behavior of Jackiw-Rebbi model is that, an in-
gap soliton state is constrained to a zero-dimensional point called as the
domain wall, and both sides of it have opposite mass terms. The simplest way
to extend a one-dimensional system is that periodically arrange a one-
dimensional system in other dimensions. Naturally, the 0D domain wall also
expends its dimensions accordingly. There are numerous relevant works to
present the properties of Jackiw-Rebbi model in two-dimensional systems. Zhang
et al. design a periodic acoustic cavity system consisting of acoustic
resonators [161] connected by thin cylindrical coupling waveguides as shown in
Fig. LABEL:JRdmintwo-dimensional(a). Its energy spectrum exhibits linear
dispersion near the $K$ and $K^{\prime}$ point, which is described by Dirac
equation (Eq. 17), named as Dirac cone. Through adjusting the height
difference $\Delta h$ of cavities, they break the inversion symmetry of the
system and successfully open a gap to introduce a topological index, valley
Chern number, obtained by integrating Berry curvature around the valley. The
opening of gap arises from the effective opposite mass terms performed around
the intersection of Dirac cone [162]. As a result, valley Chern numbers in $K$
and $K^{\prime}$ valley host opposite signs, and their signs change as the
sign of $\Delta h$ changes. When unit cells with distinct valley Hall phases
are assembled to form a domain wall, the soliton state elaborated by Jackiw-
Rebbi theory leads to gapless states propagating along the boundary, as shown
in Fig. LABEL:JRdmintwo-dimensional(b). In addition to the intuitive acoustic
cavity configuration, acoustic scatterers and mechanical structures are also
excellent platforms to realize the same physics. For example, Liu et al.
proposed two kinds of valley Hall insulators based on triangular [163] and
circular [164] (Fig. LABEL:JRdmintwo-dimensional(c)) phononic scatterers,
respectively. In the triangular scatterer structure, the valley Hall
topological transition occurs when the rotation angle of triangular scatterer
changes over $0^{\circ}$. Besides, in the circular scatterer system, it
depends on the radius difference between two kinds of circular scatterers.
Consequently, these two works observe the gapless chiral edge states
transmitting along zigzag boundary line between two domains with opposite
effective mass terms. Dispersion of edge modes is shown in Fig.
LABEL:JRdmintwo-dimensional(d). In another work, Prodan et al. establish a
mechanical counterpart based on magnetically coupled spinners [165]—stainless
steels with six arm grooves to attach neodymium disc magnet (Fig.
LABEL:JRdmintwo-dimensional(e)). And they observe the valley Hall edge modes
stemming from the interaction of soliton states localized along the domain
wall as shown in Fig. LABEL:JRdmintwo-dimensional(f). There are other methods
[39, 166, 167] to realize the Jackiw-Rebbi model in two-dimensional systems.
These above works are enough to conclude that, Jackiw-Rebbi models are
intuitively characterized by the establishment of a zero-energy soliton state
and derived hinge states in lower dimensions than that of system, even though
they usually appear at a finite frequency instead of zero energy in acoustic
lattices. These states predicted by Jackiw-Rebbi theory are one kind of low-
dimensional projections of the bulk properties termed as bulk-boundary
correspondence. This correspondence is a useful tool to characterize the
topology or polarization properties of materials.

Finally, we briefly discuss the analogues of Majorana fermions in acoustic
systems. Complex fields $\Psi$ in Dirac equation describe charged particles
and their anti-particles with opposite charges, such as electrons and
positrons. In nature, there are also some neutral particles which are their
anti-particles, as photons, phonons and gravitons etc. All of these are
integer-spin bosons, and they are described by a real field $\Psi$.

Spin-1/2 neutral particles, whose anti-particles are themselves, are termed
Majorana fermions playing an extremely vigorous role in the superconductor
realm [168]. The discussion about Majorana fermions needs the constraints of
particle-hole symmetry protecting the superconductor. In the work of Prodan et
al., they design a mechanical metamaterial [169] consisting of a periodic
array of dimers connected via two springs as shown in Fig.
LABEL:majoranainmechanic(a). For each dimer, there are two motion degrees of
freedom: one is translation and another one is rotation. The particle-hole
symmetry of this system stems from the distinct behaviors of the translational
and rotational degrees of freedom under inversion operation. Together with the
intrinsic time-reversal symmetry, the particle-hole symmetry characterizes the
BDI symmetry class. In one-dimensional fermionic topological superconductors
from BDI symmetry class, the topological edge excitations are named Majorana
fermions. Because, these edge modes are their self-mirror under the charge
conjugation, like the Majorana fermion is its own antiparticle. E. Prodan et
al. measure the topologically protected edge modes in the experiment platform
shown in Figs. LABEL:majoranainmechanic(b) and LABEL:majoranainmechanic(c),
where the domain wall is established from the exchange of the number of
springs on two diagonals for one side of unit cells. As shown in Fig.
LABEL:majoranainmechanic(b), they denote the distinct mechanical blocks with
the red and white colors on dimers. The measured topological edge modes in
Fig. LABEL:majoranainmechanic(c) are invariant under a charge conjugation. For
instance, the motion of these Majorana edge modes remains unchanged when the
translational and rotational degrees of freedom are exchanged and follow with
time-reversal operation. This is a significant feature to distinguish them
from ordinary edge modes. This fascinating work gives a method to construct
the mechanical analogues of Majorana fermions from BDI topological class,
which inspired lots of researches in this direction. What is more, the
majorana-like zero mode, reflected by the Dirac vortex originated from the
Jackiw-Rossi mechanism, has been observed in the Kekul$\acute{e}$-distorded
mechanical [170] and acoustic [171] metamaterials.

##  3 Topological phononic metamaterials

###  3.1 Valley and spin Hall phononic crystals

We first introduce the topological phononic crystals built on valley and
pseudospin degrees of freedom, in which time-reversal symmetry remains
invariant. Since these types of phononic crystals do not require time-
reversal-breaking interactions, such as measurable responses to external
magnetic fields or circulating fluid flows, they are easily achievable in
experiment, and attracted considerable attention in the early days of
exploring topological phononic crystals [172, 39, 173, 174, 163, 175, 176].
Different from the various pseudospins recently proposed to mimic the quantum
spin Hall effect [23, 177, 178], valley is a long-stand concept in solid band
theory [179], and originally introduced to indicate the local extrema
appearing in band structures. Considering that achieving topology lies chiefly
on the bands with the aggregation of Berry curvature, which is generally
inversely proportional to width of the band gap, valleys are exactly the
positions where nontrivial topology could occur. Since the wave motions in
periodic systems are marked by Bloch momentum, valleys are also used to
labeling topological states. In two-dimensional systems, a feasible way to
obtain valley states is to break the Dirac cone degeneracy [180]. The Dirac
cone in two-dimensional systems is typically represented by
$k_{x}\sigma_{x}+k_{y}\sigma_{y}$, where the Dirac velocity is set to one
without loss of generality and $\sigma_{i=x,y,z}$ are Pauli matrices. Since
the Hamiltonian only exploits two of the three Pauli matrices, to open a gap
in the Dirac cone we can introduce a perturbation term of the rest Pauli
matrix, $m\sigma_{z}$, where $m$ is mass term tuning the band gap between the
upper and lower valley states. Although the Dirac cone can appear at general
positions of the Brillouin zone without any symmetry protection [63], the
Dirac cone at the high symmetric position of the Brillouin zone is preferable
to construct the valley states, which facilitates the calculation (no need to
search the general positions in the Brillouin zone) and makes the valleys
separate in the Brillouin zone as far away as possible (leaving enough space
for regulation of the valley topology). Integration of the Berry curvature
(localized around the valley point) over one valley region returns to a non-
zero topological quantity, dubbed as valley Chern number. Since the time-
reversal symmetry is intact, an integration of the Berry curvature over a
whole band is always zero, and there always exists another valley in the same
band carrying opposite valley Chern number.

The valley phononic crystals was first built in a hexagonal lattice consisting
of regular triangular rods as its scatterers [39], where the lattice and the
scatterer have the symmetry of point groups $C_{6v}$ and $C_{3v}$,
respectively. The rotation of the scatterer [denoted by $\alpha$ in Fig.
LABEL:jiuyang_Picture1(a)] can cause a mismatch of the mirror symmetries
between the lattice and the scatterers, which leads to a reduction of the
symmetry of the phononic crystals. Specifically, the point group associated to
the corner points ($K$ and $K^{\prime}$) of the hexagonal first Brillouin zone
reduces from point group $C_{3v}$ for $\alpha=0^{\circ}$ to point group
$C_{3}$ for $\alpha=10^{\circ}$, corresponding to a transition from Dirac cone
dispersions to gapped valley dispersions [Fig. LABEL:jiuyang_Picture1(a)].
Here the valley states at the $K$ point are focused, and those at the
inequivalent K point can be deduced from the time-reversal symmetry.
Demonstrated by the distributions of the phase, $\phi$, and the time-averaged
Poynting vectors, $\langle\bm{S}\rangle\propto\bigtriangledown\phi$, of
acoustic fields (upper panels and bottom panels in Fig.
LABEL:jiuyang_Picture1(b)), the valley states have peculiar vortex features,
where the vortices are of reversed chirality for the valley states of lower
and higher frequencies ($K_{1}$ and $K_{2}$), and are spatially separated at
corners of the unit cell ($P_{1}$ and $P_{2}$). Therefore, the valley vortex
feature, originating from the one-dimensional irreducible representations of
the point group $C_{3}$, provides an obvious and measurable signal to detect
the valley states. To characterize the quality of the vortex state, a quantity
$Q=\int_{\rm{Cell}}({\bigtriangledown\times\langle{\bm{S}}\rangle})_{z}{\rm{d}}^{2}r$
can be established [39]. This quantity is proportional to the acoustic angular
momentum localized in one unit cell,
$J_{z}=\int_{\rm{Cell}}{\langle{\bf{u}}\times\rho_{0}{\bf
u^{\prime}}\rangle_{z}{\rm{d}}^{2}r}$, where $u$ is the displacement of the
vibration of acoustic field and $\rho_{0}$ is the mass density of the
background. The $J_{z}$, as well as $Q$, acts as a continuous version of the
phonon’s pseudospin angular momentum in atomic crystals [181, 182]. The
existence of nonvanishing $Q$ for the valley states implicate the intrinsic
angular momenta of the valley states and the capability of rotating objects by
transferring the angular momenta. Excitation of one single valley state in
phononic crystals is critical to possible applications in acoustic
manipulation. In Fig. LABEL:jiuyang_Picture1(c), we show that the $K$ valley
state can be excited by a Gaussian beam incidence, where the incident angle is
chosen to satisfy the transversal momentum matching. Moreover, the time-
reversal symmetry connected $K$ and $K^{\prime}$ valley states carry opposite
chirality in the same frequency, which are used to realize a valley-chirality
locked beam splitting [39, 183] in analog to the valley Hall effect [184]. As
shown in Fig. LABEL:jiuyang_Picture1(d), under excitation by a narrow Gaussian
beam, the acoustic wave in the phononic crystals spatially split into two
branches, each of which has a vortex feature with opposite chirality. Based on
this, we identify the phononic crystals with valley states reside in an
acoustic valley Hall phase.

The topologically protected acoustic edges were later observed in interfaces
between two distinct valley Hall phases [163]. The vortex chirality of the
acoustic valley states can be switched by rotating the orientation of the
triangular scatterers. Consider two phononic crystals consisting of scatterers
rotated clockwise and counterclockwise by the same angle ($\pm\alpha$), they
share the same symmetry of point group $C_{3}$ and are related by a horizontal
mirror operation, which enables them identical valley dispersions yet reverses
the chirality and position of the valley vortices. This leads to an inversion
of the valley bands, if one focuses these properties of valley vortices. The
phononic crystals thus possess two distinct valley Hall phases of opposite
mass terms, valley Hall phase-I and valley Hall phase-II, corresponding to the
rotation angles $\alpha$ belong to $(-60^{\circ},0)$ and $(0,60^{\circ})$,
respectively [163]. The valley Chern number calculated for these two valley
Hall phases are $C^{K}=1/2$ and $-1/2$, where the superscript denotes the $K$
valley [185]. As a result, the difference of $C^{K}$ across the interface
between valley Hall phase-I and valley Hall phase-II is $\pm 1$, which,
according to bulk-edge correspondence, leads to a valley induced edge state
(of positive or negative group velocity) near the projection of $K$ valley.
Similar situation occurs for the $K^{\prime}$, but $C^{K^{\prime}}=-C^{K}$ for
the $K^{\prime}$ valley [39]. Such edge states are constructed in experiment
with the setup shown in Fig. LABEL:jiuyang_Picture2(a). A pair of counter-
propagating valley edge states [green lines in the upper panel of Fig.
LABEL:jiuyang_Picture2(b)] are expected at interfaces, where the edge states
$\phi_{({\rm{I}},{\rm{II}})}^{\pm}$ and $\phi_{({\rm{II}},{\rm{I}})}^{\pm}$
respectively label the gapless modes hosted by the interfaces I-II and II-I,
along the $\pm x$ directions. Taking $\phi_{({\rm{II}},{\rm{I}})}^{+}$ as an
example, its analytical expression is proportional to a spinor $(1,1)^{T}$
with respect to the valley states ($K_{1}$ and $K_{2}$) and exponentially
decays away from the interface, which is solved by viewing valley Hall phase-I
and valley Hall phase-II as continuum media characterized by the $K$ valley
Hamiltonians with opposite mass terms [163]. The lower panel of Fig.
LABEL:jiuyang_Picture2(b) shows the measured spectra of the interface states
by Fourier transforming the scanned acoustic pressure distributions along the
interfaces, which agrees well with the simulation results. The valley edge
states can not only be supported in the horizontal interface, but also the
interface along an arbitrary direction (of angle $\theta$ with respect to the
$+x$ direction). Keeping the left- and right-hand sides of the interfaces as
valley Hall phases-II and I, the spinor factor of the edge state
$\phi_{({\rm{II}},{\rm{I}})}^{+}$ becomes $(1,e^{i\theta})^{T}$. We show a
vertical interface ($\theta=\pi/2$) in Fig. LABEL:jiuyang_Picture2(c). To
stimulate the valley edge state in that interface by an external Gaussian
beam, the incident angel should fulfill a selection rule similar to that in
Fig. LABEL:jiuyang_Picture1(c). The selection rule of excitation reflects that
the valley edge states are in nature composed of by the bulk valley states
located at $K$ or $K^{\prime}$. To validate this phenomenon, we have measured
transmissions validate the obliquely excitation. Note that a key feature of
the valley edge state is its exponential decay away from the interface, and
the decay length depends only proportionally on the mass-term difference,
i.e., a constant related to the bulk properties. As shown in bottom panel of
Fig. LABEL:jiuyang_Picture2(c), the measured transmission is indeed frequency
insensitive, in good agreement with the prediction based on the effective
Hamiltonian [163]. The negligibly weak backscattering of the valley Hall phase
edge states is demonstrated in a typical zigzag bending channel shown in Fig.
LABEL:jiuyang_Picture2(d), where the sound waves travel smoothly in the curved
path despite suffering two sharp corners. Compared with the transmission of a
straight channel of the same length, the reflection-immune edge states are
experimentally verified, as shown in Fig. LABEL:jiuyang_Picture2(e). Modifying
the scatterers, valley Hall edge states can be more localized in the interface
[186, 161] and used in acoustic directional antennas [187] and lasing modes
[188]. The valley Hall edge states have been extended to phononic crystals of
square lattice [189, 190], and further to heterostructures with Dirac
materials inserted into the interface [191].

Due to the lack of intrinsic degrees of freedom, such as spin and
polarization, for acoustic waves, exploiting the layer degree of freedom [192,
162, 193] is of great significance and fruitful in phononic crystals. This
approach becomes more feasible thanks to the developments in advanced three-
dimensional printing technology. Beyond the single layer case, bilayer valley
phononic crystals were proposed [175]. As shown in Fig.
LABEL:jiuyang_Picture3(a), the bilayer phononic crystals consists of two
layers of phononic crystals separated by a plate penetrated with a honeycomb
array of circular holes, where the orientations of the triangular rods are
characterized by the relative angle $\alpha$ and the common angle $\beta$. The
combination of these angles enables the bilayer phononic crystals with various
types of band structures, as exhibited in Fig. LABEL:jiuyang_Picture3(b) in
the vicinity of $K$ valley. The band structures, as well as their topological
properties, can be captured by an effective Hamiltonian, which, in addition to
the valley terms of single layer phononic crystals, possesses layer-dependent
terms $\eta(\alpha s_{z}+\beta)\sigma_{z}-\bigtriangleup_{c}s_{x}$ [175]. The
parameter $\eta$ depends on the spatial filling ratio of the scatterer rods,
and $\bigtriangleup_{c}$ depends on the detailed geometry of the holes
connecting the bilayers, both of which can be extracted from numerical
simulations. For fixed parameters $\eta$ and $\bigtriangleup_{c}$, all the
possible combinations of the angles $\alpha$ and $\beta$ leads to a reduced
phase diagram as shown in Fig. LABEL:jiuyang_Picture3(c). The phase diagram is
separated by the straight and curved lines associated with ring and point
degeneracies, respectively. The separated phase domains correspond to either
nontrivial valley Hall phases (shaded regions) or nontrivial acoustic layer-
valley Hall phases (unshaded regions), distinguished by the quantized
topological invariant invariants $C_{V}^{K}$ and $C_{L}^{K}$ [175]. The former
is a natural bilayer extension of the valley Chern number concerned in the
monolayer system [163, 194, 195], and the latter identifies the layer
information and resembles that proposed for quantum spin Hall systems [196,
197]. The interfaces between these topologically nontrivial phases support
acoustic edge states with different behaviors of the wave motions. Two cases
are shown in figure 3(d). One is the interface between two valley Hall phases
with the rod angles $\alpha=0^{\circ}$ and $\beta=\pm 10^{\circ}$. Crossing
the interface $\bigtriangleup C_{V}^{K}=2$ and two edge modes (red curves)
carrying positive group velocities appear near the $K$ valley, whose
eigenfields disperse in both layers and indicate a mixing of layer
pseudospins. For the interface between two layer-valley Hall phases with the
rod angles $\alpha=\pm 10^{\circ}$ and $\beta=3^{\circ}$, $\bigtriangleup
C_{L}^{K}=2$ and in contrast to the valley Hall case the two edge modes carry
different group velocities. In particular, the eigenfield concentrates
dominantly in either the upper layer or the lower layer, that is a layer-
pseudospin polarized transport. Therefore, assisted with the additional layer
information, the valley edge modes can be either layer mixed or layer
polarized, based on which even an interlayer converter has been conceived for
possible application [175]. The bilayer design late stimulates related
research in square lattice [198], and van der Waals metamaterials [199].

Another approach to increase adjustable degrees of freedom of classical waves
is to access the topologically nontrivial band gap based on a degenerate state
with higher degeneracy, such as the double Dirac cones. Following the
approach, metamaterials, analogized to quantum spin Hall effect, were proposed
for microwaves [200, 201] and Lamb waves [202], where mixed polarizations of
these vectorial fields are used to construct fermion-like pseudospins [203].
In contrast, for a scalar acoustic waves, the four-fold degeneracy of double
Dirac cones cannot be achieved by degrees of freedom of intrinsic
polarizations. To achieve the pseudospin states and the double Dirac cones in
phononic crystals, accidental degeneracies of various Bloch modes were
constructed by exploiting irreducible representations of systems with
hexagonal lattice symmetry [174, 204]. The starting point is to realize the
double Dirac cones in the Brillouin zone center via changing the filling
ratio. As shown in Fig. LABEL:jiuyang_Picture4(a), the phononic crystals is of
a graphene lattice and its filling ratio is determined by the scatterers’
radius r. At a critical filling ratio $r/a=0.3928$ ($a$ is the lattice
constant), an accidental double Dirac cone appears at the Brillouin zone
center [Fig. LABEL:jiuyang_Picture4(b)], whose dispersion can be anticipated
from an effective Hamiltonian accidentally spanned by the basis
$(p_{x},p_{y})$ and $(d_{x^{2}-y^{2}},d_{2xy})$, the two-fold irreducible
representations of the point group $C_{6v}$ [205, 206]. At a small or large
filling ratio, the double Dirac cone is gapped by two two-fold degeneracy with
the band modes being $(p_{x},p_{y})$ or $(d_{x^{2}-y^{2}},d_{2xy})$, where
pseudospins for the bulk states can be achieved through hybridizing these
modes as $p_{\pm}=p_{x}\pm ip_{y}$ and $d_{\pm}=d_{x^{2}-y^{2}}\pm id_{2xy}$
($+$ and $–$ represent the pseudospin up and down, respectively). Therefore,
by tuning the filling ratio, the energy band inversion occurs [Fig.
LABEL:jiuyang_Picture4(b)], leading to a topological transition of phononic
crystals with $p$/$d$ pseudospins. Accordingly, in the interface composed by
the phononic crystals with inverted bands, a pair of edge states is expected
to exist and traverse the overlapped band gap, shown in Fig.
LABEL:jiuyang_Picture4(c). These edge states are localized at the interface
and are recognized as $(S+iA)/(S-iA)$, where $S$ and $A$ are symmetric and
anti-symmetric modes with respect to the middle line [dashed lines in Fig.
LABEL:jiuyang_Picture4(d)]. They can also be correlated to the pseudospins of
the bulk states as $(S+iA)$/$(S-iA)=(p_{+}+d_{+})$/$(p_{-}-d_{-})$, where the
symmetric component $S$ can be represented as
$S=(p_{x}+d_{x^{2}-y^{2}})/\sqrt{2}$ and the anti-symmetric component $A$ as
$A=(p_{y}+d_{2xy})/\sqrt{2}$, as shown in Fig. LABEL:jiuyang_Picture4(d).
Hence, the two acoustic pseudospin edge states carry opposite pseudospin
feature and propagate in opposite directions, implying a one-way spin-
dependent propagation that is robust against various kinds of defect, such as
cavities, disorders, and bends [174]. In the above works, the construction of
double Dirac cones is based on the accidental degeneracy, which can be
achieved in a more deterministic way of band folding [207, 176, 208].
Specifically, as shown in Fig. LABEL:jiuyang_Picture5(a), by expanding the
primitive unit cell of phononic crystals of graphene lattice to enclose six
rods, the Brillouin zone shrinks into a new one whose area is one third of the
original Brillouin zone, and the Dirac cones locating at the corners of
original Brillouin zone are folded to the $\Gamma$ point in the new Brillouin
zone to form a double Dirac cone. To open a band gap in the double Dirac cone,
new configurations of phononic crystals are created by contracting/expanding
the metamolecule via concentrating/distracting the six rods. The gapped
dispersions are shown in Fig. LABEL:jiuyang_Picture5(b), with the pressure
field distributions of the metamolecules depicted in Fig.
LABEL:jiuyang_Picture5(c). Owing to a multiple scattering inside the
metamolecule, a pair of dipolar resonant states accompanied by a pair of
quadrupolar resonant states are created, echoing the basis $p_{x/y}$ and
$d_{(x^{2}-y^{2})/{2xy}}$. Again, the band inversion caused by contracting and
expanding the rods in one unit cell leads to counterpropagations of edge
states with opposite pseudospin features. Note that the six-fold rotation
symmetry plays an important role in the design of phononic crystals, as it
allows two inequivalent two-dimensional modes as the cornerstones to build the
pseudospins. With similar design schemes, acoustic analogues of quantum spin
Hall effect have been realized in mechanical waves [209, 210], mechanical
systems [211], and recently extended to three-dimensional phononic crystals by
layer-stacking structures [212, 213].

###  3.2 Acoustic Chern insulator and spin-Chern insulator

According to the time-reversal symmetry, the two-dimensional topological
insulators usually come in two different classes. The first breaks the time-
reversal symmetry, and hosts the gapless chiral edge states. It hosts the
quantum anomalous Hall effect and is commonly called Chern insulator, since
its topological invariant is described by Chern number [41, 66]. For acoustic
waves, circulating fluid flow is introduced to break the time-reversal
symmetry [214]. Inspired by this, the acoustic Chern insulator is proposed in
the acoustic metamaterial containing circulating fluids [215, 216, 217], and
later realized in the resonator array with the chiral-structural rotors [218].
Figure LABEL:weiyin.1.CI_and_SCI(a) shows the sample, i.e., a honeycomb
lattice of ring resonators. Each resonator has a rotor rotating by a motor,
generating a stable air flow and breaking the time-reversal symmetry, as shown
in Fig. LABEL:weiyin.1.CI_and_SCI(b). At this time, the bulk band hosts
nontrivial topology with nonzero Chern number, and gives rise to the chiral
edge states for an open boundary, as calculated in Fig.
LABEL:weiyin.1.CI_and_SCI(c). The measured pressure fields in Fig.
LABEL:weiyin.1.CI_and_SCI(d) show the anticlockwise chiral edge mode can
transport across a defect where the circulator stops motion, demonstrating the
robustness of the edge states.

The second class of two-dimensional topological insulator keeps the time-
reversal symmetry, and hosts a pair of helical edge states. It hosts the
quantum spin Hall effect in the electronic systems [23, 177], whose bulk band
topology can be described by the $Z_{2}$ index [43] or the spin-Chern numbers
equivalently [197, 219]. The gapless feature of the helical edge states is
guaranteed by the Kramer degeneracy originated from the spin-1/2 time-reversal
symmetry. Different from the $Z_{2}$ index, spin-Chern numbers are well
defined in the absence of any symmetries [220], thus are employed to identify
pseudospin topological insulators [221], forming the concept of spin-Chern
insulator [219, 220, 221, 222, 223, 224, 225]. Whether the helical edge states
in spin-Chern insulator are gapless or not, relates with the symmetry of
system and the microstructure of sample boundary [224]. Figure
LABEL:weiyin.1.CI_and_SCI(e) shows the sample of the acoustic spin-Chern
insulator [225], which is a bilayer structure with a layer pseudospin degree
of freedom. As shown in Fig. LABEL:weiyin.1.CI_and_SCI(f), the chiral
interlayer tubes induce the pseudospin-orbit coupling, giving rise to the
nontrivial band topology described by nonzero spin-Chern numbers. The gapless
helical edge states are calculated and observed in Fig.
LABEL:weiyin.1.CI_and_SCI(g). Owning to the pseudospin-momentum locking, they
are robust against the backscattering induced by the defect, as shown in Fig.
LABEL:weiyin.1.CI_and_SCI(h).

###  3.3 Weyl acoustic metamaterials

Different from the topological insulators, topological semimetals are featured
with band touching in momentum space, including zero-dimensional discrete
points, one-dimensional continuum lines or two-dimensional surfaces [29, 226].
The band touching carries nontrivial topology and gives rise to boundary
states. Conventional Weyl semimetals are crystals hosting type-I Weyl
points—the twofold linear crossing points without overtilt in momentum space.
The quasiparticle excitations around the type-I Weyl point satisfy the Weyl
equation, thus behaves like Weyl fermion. The realization of Weyl fermions in
lattice systems have been generalized from electronic systems to atomic
systems (see for example [227, 228]) and then to photonic (see for example
[229, 230, 231]) and phononic systems. In phononic crystal, the conventional
Weyl semimetal was proposed [232] and realized [233, 234] by a layer-stacking
construction. The linear dispersion in two dimensions is easily obtained in
graphene. Stacking the graphene along the $z$ direction with chiral interlayer
coupling, as shown in Fig. LABEL:weiyin.3.type-I_Weyl(a), the linear
dispersion also occurs in the $z$ direction, giving rise to the Weyl point.
The effective Hamiltonian of Weyl point has the form as
$H_{\text{W}}=A_{x}k_{x}\sigma_{x}+A_{y}k_{y}\sigma_{y}+A_{z}k_{z}\sigma_{z}$,
where $\bm{k}$ is the wavevector, $\bm{\sigma}$ is the Pauli matrix, and
$A_{i}$ with $i=x,y,z$ is the coefficient. Figure LABEL:weiyin.3.type-
I_Weyl(b) shows the corresponding unit cell of the acoustic metamaterial.
There exist four Weyl points in the first Brillouin zone, as shown in Fig.
LABEL:weiyin.3.type-I_Weyl(c). The linear dispersion along the $k_{z}$
direction is calculated in Fig. LABEL:weiyin.3.type-I_Weyl(d). Robustness of
the Weyl points has two aspects: the first is that the Weyl point is stable
against weak perturbations owning to describing by all the three Pauli
matrices; the second is that it hosts nontrivial topological charge described
by Chern number $C=\text{sgn}(A_{x}A_{y}A_{z})=\pm{1}$, behaving as the drain
or source of Berry flux.

Owning to their topological nature, Weyl points always come in pairs with the
zero summing charges, giving rise to Fermi-arc surface states and chiral
anomaly. Unlike the Fermi-circle surface states in topological insulators, the
Fermi-arc surface states are nonclosed and connect the Weyl points with
opposite charges. Figures LABEL:weiyin.3.type-I_Weyl(e) and
LABEL:weiyin.3.type-I_Weyl(f) show the sample and unit cell of the Weyl
acoustic metamaterial in experiment, respectively [233]. Fermi arcs,
equifrequency contours of the surface dispersion, are observed in Fig.
LABEL:weiyin.3.type-I_Weyl(g). The Fermi-arc surface states can also be
revealed by the surface dispersions for fixed $k_{z}$, as shown in Fig.
LABEL:weiyin.3.type-I_Weyl(h). These surface dispersions host the chiral
surface states, attributed to the $k_{z}$-dependent Chern numbers, which are
related with the charges of Weyl points. The equifrequency cut of the surface
dispersions forms a trajectory connected a pair of Weyl points with opposite
charges, i.e., Fermi arc. As such, the Fermi-arc surface states in the Weyl
acoustic metamaterial exhibit topologically protected one-way propagation of
acoustic waves in the presence of defects.

The Fermi-arc surface states can give rise to topological negative refraction
[235], providing a practical use of Fermi arcs. Figure
LABEL:weiyin.4.Fermi_arc_and_LL(a) shows the Fermi arcs of two neighboring
surfaces, which form an interface (black vertical line). Considering an
incident wave along the direction of the red arrow (labelled 1), the refracted
wave will propagate along the direction of the blue arrow (labelled 3), since
the wavevector is conserved parallel to the interface. As shown in Fig.
LABEL:weiyin.4.Fermi_arc_and_LL(b), this refraction is negative, because the
refracted and incident waves are on the same side of the normal (dashed
horizon line). More importantly, owing to the nonclosed nature of Fermi arcs,
there are no modes for reflected wave, thus the reflection (dashed arrow
labelled 2) is forbidden. Figure LABEL:weiyin.4.Fermi_arc_and_LL(c) shows the
measured pressure fields on the surface of the Weyl acoustic metamaterial,
evidencing the negative refraction without reflection of the Fermi-arc surface
states.

Under a strong magnetic field, the Weyl point supports the chiral zeroth
Landau level, which hosts one-way propagation related with the charge of Weyl
point [236]. While adding a parallel electric field, the Weyl quasiparticles
can be pumped through the zeroth Landau level between the Weyl points with
opposite charges, leading to non-conservation of chiral currents, i.e., chiral
anomaly, the other hallmark of Weyl semimetal. Since the acoustic wave is
inert to the external magnetic field, constructing pseudomagnetic field is
significant in acoustic metamaterials. By designing an effective space-
dependent sublattice potential along the $x$ direction, the Weyl points are
shifted along the $k_{z}$ or $-k_{z}$ direction, as shown in Fig.
LABEL:weiyin.4.Fermi_arc_and_LL(d), axial pseudomagnetic fields are induced
along the y direction in the Weyl acoustic metamaterial. Weyl points at $H$
and $K^{\prime}$ feel an opposite field to the ones at $H^{\prime}$ and $K$.
Unlike the genuine magnetic field, the acoustic metamaterial keeps the time-
reversal symmetry in the presence of the pseudomagnetic fields, giving rise to
the chiral zeroth Landau levels illustrated in Fig.
LABEL:weiyin.4.Fermi_arc_and_LL(e). For example, due to the pseudomagnetic
fields with opposite directions acting on Weyl points at $H$ and $H^{\prime}$,
the chiral Landau levels around them have opposite group velocities, although
they host the same charge +1, as observed in Fig.
LABEL:weiyin.4.Fermi_arc_and_LL(f).

###  3.4 Other Weyl acoustic metamaterials

Unlike the Weyl fermions in high-energy physics, the quasiparticle excitations
in crystals are not limited by the Poincaré symmetry, thus exhibit rich
configurations with nontrivial topology in momentum space. The typical one is
the type-II Weyl point, characterized by an overtilted cone dispersion [237].
Although the topological charge of Weyl point is independent on the overtilted
dispersion, this feature gives rise to unique properties different from the
type-I Weyl point, such as the antichiral zeroth Landau level and anisotropic
chiral anomaly. The type-II Weyl points have been predicted in the acoustic
metamaterials with stacked hexagon and square structures [232, 31], and later
observed in experiments. Figures LABEL:weiyin.5.type-II_Weyl(a) and
LABEL:weiyin.5.type-II_Weyl(b) show the unit cell and the observations of the
type-II Weyl points in an acoustic metamaterial, respectively [238]. Due to
lacking the relation by symmetry, the Weyl points are at different
frequencies. Then the ideal type-II Weyl points, which are related by the
mirror and time-reversal symmetries and thus reside at the same energy, have
been realized in the acoustic metamaterial [239], as seen in Figs.
LABEL:weiyin.5.type-II_Weyl(c) and LABEL:weiyin.5.type-II_Weyl(d). By
constructing the pseudomagnetic fields in this ideal type-II Weyl acoustic
metamaterial, the antichiral zeroth Landau levels were observed [240].

The second way to generalize the conventional Weyl point is changing the
topological charge of the twofold degenerate point, which is no longer the
linear dispersions in all directions, such as the quadratic and quadruple Weyl
points [241, 242, 243]. The Weyl point possessing high charge is usually
protected by the spatial symmetries, to vanish the linear dispersion. Figure
LABEL:weiyin.6.high_charge_Weyl(a) shows the sample of acoustic metamaterial
possessing a pair of quadratic Weyl points located at $\Gamma$ and A, together
with the conventional Weyl points [242]. Quadratic Weyl points host charge
$\pm{2}$, and featured with the quadratic dispersions in two directions and
linear dispersion in the other one. Figure LABEL:weiyin.6.high_charge_Weyl(b)
shows the measured projected bulk band dispersion, exhibiting the nonlinear
feature of the quadratic Weyl point. Two chiral surface states with positive
group velocity in the surface dispersion along a loop centered at
$\overline{\Gamma}$, as shown in Fig. LABEL:weiyin.6.high_charge_Weyl(c),
demonstrates the quadratic Weyl point possesses charge +2 at $\Gamma$. The
quadruple Weyl point, which carries the highest charge of a twofold degenerate
point with charge 4, has also been observed in an acoustic metamaterial with
space group symmetry $P$432 [243]. Figures LABEL:weiyin.6.high_charge_Weyl(d)
and LABEL:weiyin.6.high_charge_Weyl(e) show the sample and the bulk
dispersion. The quadruple Weyl point exhibits the cubic dispersion in the
(111) direction and quadratic dispersions in the other two directions.
Similarly, as shown in Fig. LABEL:weiyin.6.high_charge_Weyl(f), four chiral
surface states with positive group velocity reveal the charge 4 of the
quadruple Weyl point.

The third way is changing the low-energy excitation from the spin-1/2
quasiparticle to the other spin one, accompanied by changing the degeneracy of
the touching point. Spin-1 Weyl point is the typical one, which is the
threefold degenerate point touched by two linear dispersions and an additional
flat band, and carries charges ($-2$,0,2) or (2,0,$-2$) for the three bulk
bands. The low-energy excitation has the form $\bm{k}\cdot\bm{S}$, where
$\bm{S}$ is the vector of the spin-1 Gell-Mann matrix, rather than the
spin-1/2 Pauli matrix, thus behaves as the spin-1 quasiparticle. Due to
describing by only part of Gell-Mann matrices, spin-1 Weyl point is usually
protected by external symmetries. Figure LABEL:weiyin.7.spin-1_Weyl(a) is the
unit cell of an acoustic metamaterial with space group symmetry $P$213 [244].
As shown in Fig. LABEL:weiyin.7.spin-1_Weyl(b), the acoustic metamaterial
possesses a spin-1 Weyl point coexisting with a charge-2 Dirac point. Charge 2
of these two points leads to the double Fermi arcs in the surface Brillouin
zone, as calculated and measured in Figs. LABEL:weiyin.7.spin-1_Weyl(c) and
LABEL:weiyin.7.spin-1_Weyl(d), respectively. The spin-1 Weyl semimetal
exclusively possessing spin-1 Weyl points can be realized in an acoustic
metamaterial by stacking the Lieb lattice layer [245], with the unit cell
given in Fig. LABEL:weiyin.7.spin-1_Weyl(e). A pair of spin-1 Weyl points, as
shown by the bulk dispersion in Fig. LABEL:weiyin.7.spin-1_Weyl(f), are
connected by the chiral symmetries. Charges ($-2$,0,2) and (2,0,$-2$) for the
three bulk bands result in the surface states shown in Fig.
LABEL:weiyin.7.spin-1_Weyl(g) and double Fermi arcs in Fig.
LABEL:weiyin.7.spin-1_Weyl(h) emerging between the first and second bands, as
well as between the second and third bands. Along this generalization,
spin-3/2 and spin-5/2 quasiparticles can be explored in acoustic
metamaterials.

Recently, the higher-order Weyl semimetals are realized in acoustic
metamaterials. As a significant extension, the higher-order Weyl point not
only hosts the nonzero topological charge, as the first-order feature, but
also can be viewed as a transition point between second-order topological
insulator and Chern insulator or topologically trivial insulator [246, 62].
Besides the two-dimensional Fermi-arc surface states, higher-order Weyl
semimetal exhibits the one-dimensional hinge states, connecting the higher-
order Weyl points. The higher-order Weyl semimetal was realized in an acoustic
metamaterial by stacking the kogome lattice layer [247], whose unit cell is
shown in Fig. LABEL:weiyin.8.higher-order_Weyl(a). This system hosts the pure
higher-order Weyl points of four, showing in Fig. LABEL:weiyin.8.higher-
order_Weyl(b). In this case, the Chern number as a function of $k_{z}$ is
always zero, but the bulk polarization as the higher-order topological
invariant has distribution, as shown in Fig. LABEL:weiyin.8.higher-
order_Weyl(c). The nonzero $k_{z}$-dependent bulk polarizations lead to the
hinge states, as measured and calculated in Fig. LABEL:weiyin.8.higher-
order_Weyl(d). The higher-order Weyl semimetal can also be realized in an
acoustic metamaterial with a chiral screw symmetry $S_{4z}$ [248]. Figure
LABEL:weiyin.8.higher-order_Weyl(e) shows the unit cell. The acoustic
metamaterial hosts two higher-order Weyl points with charge $-$1 and a
quadratic Weyl point with charge +2, as depicted in Fig.
LABEL:weiyin.8.higher-order_Weyl(f). The topological property of the acoustic
metamaterial can be revealed by the two-dimensional topological index, Chern
number as the first order topological invariant and quadrupole index, by
considering $k_{z}$ as a parameter. As shown in Fig. LABEL:weiyin.8.higher-
order_Weyl(g), the transition points both for the Chern number and quadrupole
index are the higher-order Weyl points. Figure LABEL:weiyin.8.higher-
order_Weyl(h) shows the measured pressure fields, demonstrating the hinge
states, owning to the higher-order topology.

###  3.5 Dirac acoustic metamaterials

Conventional three-dimensional Dirac point is a fourfold linear crossing
point, in which the quasiparticle excitation behaves as Dirac fermion. The
Dirac point is constituted by a pair of Weyl points with opposite charges,
thus carries a charge described by $Z_{2}$ or spin-Chern number. The
accidental Dirac point can be formed by band inversion [249], while the
symmetry-enforced Dirac point usually emerges in the acoustic metamaterial
with nonsymmorphic space group [250, 251]. Figure LABEL:weiyin.9.Dirac(a)
shows the unit cell of an acoustic metamaterial with space group symmetry
$Ia\overline{3}$, which hosts a pair of Dirac points [250, 251]. Figure
LABEL:weiyin.9.Dirac(b) is the calculated bulk dispersion, showing a Dirac
point at P. The other one connected by the symmetry is at $-$P. Although the
Fermi-arc surface states are usually not stable in Dirac semimetal [252], they
are robust here protected by the combination of gride and time-reversal
symmetries, as shown in Fig. LABEL:weiyin.9.Dirac(c). Naturally, there exist
the higher-order Dirac points, which also have been explored in acoustic
metamaterials [253, 254]. Figures LABEL:weiyin.9.Dirac(d) and
LABEL:weiyin.9.Dirac(e) show the sample and the bulk dispersion of a higher-
order Dirac acoustic metamaterial, respectively. Similar to the Weyl case, the
higher-order topology here can be revealed by the two-dimensional higher-order
topological index, as calculated in the inset of Fig. LABEL:weiyin.9.Dirac(e).
Nontrivial higher-order topology gives rise to the hinge dispersion, as shown
in Fig. LABEL:weiyin.9.Dirac(f). In addition, there exist charge-2 Dirac point
observed in acoustic metamaterials, which is constituted by two Weyl points
with same charge [244].

###  3.6 Nodal line and nodal surface acoustic metamaterials

Nodal line semimetal hosts the band touching of the one-dimensional continuum
lines. Different from the nodal point possessing nonzero charge described by
two-dimensional topological invariant, such as Chern number for Weyl point,
nodal line usually carries nontrivial $\pi$ Zak phase, one-dimensional
topological invariant, and give rise to the drumhead surface states [255]. The
realization of nodal line in phononic system was first investigated
theoretically by Xiong et al. [256] where nonsymmorphic crystalline symmetry
is exploited to generate symmetry-protected, unavoidable phononic nodal lines.
Later, phononic nodal lines were investigated in various platforms. Figure
LABEL:weiyin.10.line_and_surface(a) shows the acoustic metamaterial sample
hosting a nodal ring in the first Brillouin zone [257], which is protected by
a mirror symmetry. Figure LABEL:weiyin.10.line_and_surface(b) shows the
distribution of nodal ring in reciprocal space. Figure
LABEL:weiyin.10.line_and_surface(c) is the measured equifrequency contour,
evidencing the existence of the nodal ring. By parameterizing $k_{x}$ and
$k_{y}$, the Zak phase calculated in $k_{z}$ direction equals to $\pi$ or $0$,
transiting at the projected nodal rings. Considering an open boundary along
the $z$ direction, the end states resulted from the $\pi$ Zak phase form the
flat drumhead surface states, as shown in Fig.
LABEL:weiyin.10.line_and_surface(d). Other geometric configurations, such as
straight line [258] and chain [259], have also been realized in acoustic
metamaterials.

When the band touching is the two-dimensional surface, i.e., nodal surface,
the topological property returns to the case of nodal point [260, 261]. Nodal
surface carries a nonzero topological charge, thus behaves like the nodal
point with the same charge, similar to the nodal line with a nonzero
topological charge. Figure LABEL:weiyin.10.line_and_surface(e) shows the unit
cell of an acoustic metamaterial with the combination of time-reversal
symmetry and a twofold screw symmetry along the $z$ direction. As shown in
Fig. LABEL:weiyin.10.line_and_surface(f), the system hosts a nodal surface
with charge $-2$ and two Weyl points with the same charge +1. The bulk
dispersion exhibiting the nodal surface is calculated in Fig.
LABEL:weiyin.10.line_and_surface(g). Two Fermi arcs connected the nodal
surface and Weyl points are measured in Fig.
LABEL:weiyin.10.line_and_surface(h), as expected. Along this way, the nodal
point may be extended to a nodal line degeneracy carrying the same charge in
an acoustic metamaterial [262].

###  3.7 Higher-order topological insulators

Since 2018, an intriguing class of topological insulating phases called
higher-order topological insulators, which go beyond the conventional bulk-
edge correspondence, have attracted tremendous attention [263, 264, 265, 266,
267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 53, 278, 279, 280, 281,
282, 283, 284, 285, 34, 286, 287, 288, 36, 289, 290, 291, 292]. In general, a
$l$th order topological insulator in $n$ dimensional systems has topologically
protected boundary states on the $n-l$ dimensional boundaries. For example, a
two-dimensional higher-order topological insulator hosts gapped one-
dimensional edge states and zero-dimensional corner states in the band gap. In
this section, we discuss the recent progresses on phononic higher-order
topological insulators.

The study of higher-order topological insulators stems from several aspects.
First, with the progressive understanding of topological crystalline
insulators (i.e., topological insulating phases protected by crystalline
symmetries), it became clear that as the symmetry of the edge boundaries is
lower than the bulk, the predicted topological edge states often become
gapped—similar to the gapped edge states in electronic $Z_{2}$ topological
insulators when the time-reversal symmetry is broken [293]. Surprisingly, the
gapped edge states may by themselves become effectively lower-dimensional
topological insulators and may support topologically protected, nontrivial
corner or hinge states, yielding a dimensional hierarchy of topological
boundary states [40, 213, 265, 266, 267, 268, 269, 294, 271, 272, 273, 274,
275, 276, 277]. Figure LABEL:haixiao.TCI(a) presents schematically an example:
a two-dimensional higher-order topological insulator hosts both the gapped
edge states and in-gap corner states. The gapped edge states originate from
the bulk topological crystalline phase and the breaking of the protective
crystalline symmetry at the edge boundaries. These gapped edge states can be
described by massive one-dimensional Dirac equations. The switch of the sign
of the Dirac mass at the adjacent edge boundaries then leads to the
topological corner states according to the Jackiw-Rebbi mechanism.

A concrete example of the acoustic realization of such a two-dimensional
higher-order topological insulator is given in Fig. LABEL:haixiao.TCI(b). In
this acoustic metamaterials, the topological transitions of the bulk and edges
can be triggered independently by tuning a single geometric parameter
$\theta$. To visualize the corner mode, a box-shaped supercell consisting of
acoustic metamaterials with a distinct higher-order topology is constructed,
of which the field pattern is displayed in Fig. LABEL:haixiao.TCI(c). Such a
proposal opening a new route towards tunable topological metamaterials and
subsequently has been realized in the mechanical system [274, 273]. Another
common Jackiew-Rebbi mechanism induced higher-order topological insulator is
the acoustic topological crystalline insulator (also known as the acoustic
analog of the quantum spin-Hall insulator), in which the bulk topological
index and the bulk-induced corner topological index can be defined via the
symmetry-eigenvalues of Bloch functions at the high symmetry points [295].
When the acoustic topological crystalline insulator and normal insulator are
placed in a supercell with specific symmetry, the nontrivial corner index
difference gives rise to the corner states featured with pseudospin-momentum
locking, leading to the higher-order topological quantum spin-Hall effect
[266, 294, 271]. In addition, to overcome the shortage that the inducing
topological corner mode emerges in very limited geometries, a general scheme
based on Dirac vortices from aperiodic Kekulé modulation is also proposed
[296]. By further extending to three dimensions, one may realize the
hierarchies of topological insulators [213, 272, 267, 275] and Dirac points
[268, 269]. For example, a three-dimensional acoustic analogy of topological
insulator exhibits both two-dimensioanl gapless surface Dirac cone as first-
order and one-dimensional gapless hinge Dirac dispersion as second-order
topological insulators, supporting robust surface or hinge sound transport
[213]. Such an idea can be further generalized to realize hinge propagation in
three different directions [272, 267, 275]. As depicted in Fig.
LABEL:haixiao.TCI(d), a three-dimensional sample consisting of three-
dimensional topological phononic crystals is fabricated to illustrate the
hinge states transporting along three different directions. By carefully
designing onsite potential and the intracell and intercell layer coupling,
different topological phases can be obtained in the stacked bilayer hexagonal
lattice, which further enables the interface states. As shown in Figs.
LABEL:haixiao.TCI(e) and LABEL:haixiao.TCI(f), the experimentally measured
projected band dispersion along the $k_{x}$ and $k_{z}$ direction are
observed, indicating the hinge transport along $x$ and $z$ directions,
respectively. By further printing a sample that consisting of different
phases, the one-way transport from hinge to hinge is demonstrated, of which a
measured acoustic pressure field distribution is displayed in Fig.
LABEL:haixiao.TCI(g).

The second origin of higher-order topological insulators is the polarization
and Wannier center of crystalline insulators. In the modern theory of
polarization, the bulk dipole moment is formulated through the Berry phase,
which is related to the Wannier center. The dipole moment can be quantized by
crystalline symmetries, such as mirror and rotation symmetries, making it
eligible as a topological invariant. A typical example with quantized dipole
moment is given by the well-known Su-Schrieffer-Heeger model [see the upper
panel of Fig. LABEL:haixiao.Wannier_HOTI(a)], which describes one-dimensional
spinless electronic system in a chain with staggered hopping amplitudes [21].
When Su-Schrieffier-Heeger model has a nontrivial dipole moment, namely, the
Wannier center configuration of the Su-Schrieffier-Heeger model differs from
its lattice site configuration, the fractional charges emerge at boundaries
that cannot be removed by any perturbations, as depicted in the lower panel of
Fig. LABEL:haixiao.Wannier_HOTI(a). Although the above polarization theory was
first proposed in solid physics, it still can be applied to the understanding
of topological behavior in acoustics. Note that the dipole polarization should
be replaced by the concept of the Wannier center, and the fractional boundary
charge can be understood as due to the filling anomaly of the bulk and its
manifestation at the boundaries [297]. We remark that this picture holds not
only in the tight-binding models but also in phononic systems which cannot be
described by tight-binding models, as summarized in Ref. [292].

Inspired by the Su-Schrieffer-Heeger model, the first demonstration of
nontrivial topology in acoustics was realized in 1D phononic crystals [32]. An
interface state emerges at the interface when two phononic crystals with
different dipole moments are placed together [see Fig.
LABEL:haixiao.Wannier_HOTI(b)]. Starting from the quantized dipole moment in
one-dimensional systems, one can generalize it to higher-dimensional systems,
of which have polarization forms similar to that of the dipole moment in one
dimension. In the literature, topological crystalline insulators characterized
by nontrivial dipole momentum are termed Wannier-type higher-order topological
insulators [53, 278, 279, 280, 281, 282, 283, 298, 299, 284, 285, 300, 291,
301, 302, 303]. A prototype of Wannier-type higher-order topological
insulators is the kagome lattice with dimerized couplings[see Fig.
LABEL:haixiao.Wannier_HOTI(c,d)], which theoretically proposed in a tight-
binding model [53] and subsequently experimentally demonstrated in acoustic
systems [278, 279][also see Fig. LABEL:haixiao.Wannier_HOTI(e,f)]. These
topological phases are characterized by a topological invariant, defined as an
integral of the Berry potential over the entire Brillouin zone. Since the
dipole polarization is equivalent to the position of Wannier center [see Fig.
LABEL:haixiao.Wannier_HOTI(c)], it is intuitively to distinguish the
topological phase by depicting the Wannier center configuration [280]. The
mismatch between the lattice site and Wannier center gives rise to the higher-
order topological corner states. By combining the valley degree of freedom,
the emergence of corner states is tied to specific corners[see Fig.
LABEL:haixiao.Wannier_HOTI(g-i)], results in the valley-selective corner
states [281]. To step further, one can extend the two-dimensional second-order
topological insulator to the three-dimensional third-order topological
insulator [see the tight-binding model in Fig. LABEL:haixiao.Wannier_HOTI(g)].
As shown in Fig. LABEL:haixiao.Wannier_HOTI(h), an anisotropic diamond lattice
with cubic unit cell is proposed to mimic the third-order Wannier-type
topological insulators. Experimentally, a third-order topological insulator
with rhombohedronlike structure containing six rhombus-shaped surfaces [see
Fig. LABEL:haixiao.Wannier_HOTI(i)] is carefully designed by stereo-
lithography three dimension printing [282]. Since a third-order topological
insulator is characterized by the existence of topological corner states at
certain corners of a finite three-dimension sample, Fig.
LABEL:haixiao.Wannier_HOTI(j) also presents a typical corner state via
measuring the local acoustic response at two corner resonators. Other schemes
to third-order topological insulators also can be accessed by the acoustic
pyrochlore lattice [300] and an Su-Schrieffer-Heeger-inspired cubic lattice
[291, 301].

Although the dimensional hierarchy of the Jackiw-Rebbi mechanisms is an easy
way to understand and construct higher-order topological insulators,
historically, higher-order topological insulators were first found in the
study of the multipole moment in crystalline insulators. In the modern theory
of polarization, the bulk dipole moment is formulated through the Berry phase.
However, the generalization to multipole moment leads more intricate
formulation. In contrast to the Wannier-type higher-order topological
insulators, multipole topological insulators have vanishing dipole moment but
with quantized multipole moments [264, 297, 34, 286, 36, 289] that are
described by nested Wannier bands [264, 297, 288, 287] and quantum operator
formulations [304, 305, 306]. For example, a quadruple topological insulator
has vanishing dipole polarization but a finite quadrupole moment that
protected by certain symmetries, as depicted in Fig.
LABEL:haixiao.3.multipole_HOTI(a). Fig. LABEL:haixiao.3.multipole_HOTI(b)
presents the tight-binding model for quadrupole topological insulators, which
is a square lattice model with four sites in each unit cell with coexisting
both positive and negative nearest-neighbor couplings [264, 297]. The positive
and negative couplings enable the $\pi-$flux per plaquette (the area of one
unit cell), leading to two noncommutative reflection symmetries, which is
necessary for emergency nontrivial quadrupole topology. To this end, a
symmetry-based approach is introduced to realize quadrupole topological
insulator [286], in which the quadrupole topology is protected by two
noncommutative glide symmetries (see Fig. LABEL:haixiao.3.multipole_HOTI(c)).
Another approach to acoustic quadrupole topological insulator is to generate
both positive and negative hoppings directly by linking acoustic cavities with
different connectivity according to the field morphologies of acoustic
resonators [36] (also see Fig. LABEL:haixiao.3.multipole_HOTI(d)).

Similar to the quadrupole topological insulator with quantized quadrupole
moment, an octupole insulator has vanishing dipole polarization and quadrupole
moment but a finite octupole moment (see Fig.
LABEL:haixiao.3.multipole_HOTI(e)). The tight-binding model for octupole
topological insulator use cubic lattice models with eight sites in each cell
with coexisting positive and negative nearest shopping [also see Fig.
LABEL:haixiao.3.multipole_HOTI(f)] [264, 297]. Due to its complex tight-
binding configurations, the three-dimensional octupole topology is even more
challenging to be realized. Up till now, the only way to acoustic octupole
topological insulator is to generate the positive and negative hoppings by
linking resonators with different configurations [289, 307]. A typical
fabricated sample is given in Fig. LABEL:haixiao.3.multipole_HOTI(g). The
manifestation of the octupole topological insulator can be identified by the
emergence of the surface, edge, and corner states, respectively. As shown in
Fig. LABEL:haixiao.3.multipole_HOTI(h), the measured acoustic frequency-
response spectra for the selective bulk (orange), surface (yellow), edge
(blue) and corner sites (red) of a finite structure provide the direct
evidence for the acoustic octupole topological insulators.

Besides, it is also worthy to notice several new concepts that are related to
the higher-order topological insulators. For example, by utilizing the
richness classes of higher-order topologies, one may combine the conventional
topology with higher-order topological phase in a single acoustic system,
leading to the hybrid (or hierarchy) topological insulator [290, 291]. In
addition, an acoustic square-root topological insulator, whose topological
property is inherited from the square of the Hamiltonian, is also demonstrated
to host higher-order topology [308]. Last but not least, the gain-and-loss-
induced higher-order topological insulator is also proposed to be implemented
in sonic crystals [309]. Subsequently, a higher-order topological insulator
induced by deliberately introduced losses was experimentally realized in a
phononic crystal [37].

###  3.8 Non-Abelian topological states

In this section, a brief introduction to the non-Abelian band topology and a
short review of its recent experimental realizations in phononic metamaterials
are presented. The non-Abelian band topology, which was first proposed by Wu
et al. in 2019 [310], to some degree, complements and extends the conventional
Abelian topological band theory. Let us first recall the Abelian band topology
before we move on. As elucidated in other sections, the topological nature of
band insulators, or degenerate band nodes, in general, can be characterized
by, e.g., the Chern number, quantized Zak phase, and various winding numbers,
which are usually defined within two subspaces spanned by “occupied” and
“unoccupied” bands. These topological invariants or node charges are
classified by Abelian groups so that they obey the additivity and
commutativity of multiplication. A prototype example is that two Weyl points
with opposite Chern charges tend to annihilate in collision.

Beyond the above paradigm, the non-Abelian band topology, as its name implies,
exemplifies the non-Abelian topological charges of multiple bands, wherein the
emergence and robustness of multi-band configurations cannot be explained by
the conventional Abelian topological invariants. So far, the application of
the non-Abelian band topology always demands two pivotal elements. First, the
Hamiltonian of the system should enjoy the space-time inversion symmetry
$I_{ST}=I_{S}T$, where $I_{S}$ denotes the spatial inversion $P$ (or $\pi$
rotation in two-dimensional spinless systems) and $T$ the time inversion.
Second, multiple bands or gaps require to be taken into consideration
simultaneously, rather than to be simply partitioned into two subspaces. The
combined anti-unitary symmetry $I_{ST}$ acts as an on-site symmetry since it
leaves every $k$ point fixed in momentum space. More importantly,
$I_{ST}^{2}=1$ leads us to take a suitable basis so that $I_{ST}$ can be
represented by the complex conjugation $K$, which yields

$I_{ST}H(k)I_{ST}^{-1}=H^{*}(k)=H(k);\quad
I_{ST}|u(k)\rangle=|u(k)\rangle^{*}=|u(k)\rangle.$ (18)

In that case, the Hamiltonian as well as the corresponding eigenstates at each
$k$ point is real-valued. The gauge freedom of eigenstates is reduced to $\pm
1$. The Hamiltonian at each $k$ is then encoded by an orthonormal rotation
frame $\\{u_{i}(k)\\}$, which denotes a local geometric quantity of
eigenstates. The way how the frame smoothly rotates when the Hamiltonian
evolves with $k$ contributes to the band topology in $I_{ST}$-symmetric
systems. For example, the first Stiefel-Whitney number defined from the twist
of real eigenstates is equivalent to the quantized Zak phase derived from the
parallel transport based on complex eigenstates [311].

The power of the multi-gap perspective is released when the order–parameter
space of the Hamiltonian is involved. As illustrated in Fig. LABEL:figure1,
when we consider a $PT$-symmetric three-band spinless system, if we separate
three bands into one occupied and two unoccupied bands, then the corresponding
order–parameter space of the Hamiltonian, without degeneracy, is
$M_{1,2}=O(3)/O(1)\times O(2)$, which is a real Grassmann manifold where
$O(N)$ is the $N$-dimensional orthogonal group [312]. According to the
homotopy group theory, the first homotopy group of the real Grassmann manifold
$M_{1,2}$ is $\pi_{1}(M_{1,2})=Z_{2}$, which accounts for the $Z_{2}$
quantized Zak phase used in Abelian band topology [313, 314, 315].

In comparison, when we consider the multiple gaps together (see the right
panel in Fig. LABEL:figure1), the order-parameter space of the Hamiltonian is
then altered to a real flag manifold [312]

$M_{3}=O(3)/O(1)\times O(1)\times O(1).$ (19)

The first homotopy group of the real flag manifold $M_{3}$ is

$\pi_{1}(M_{3})=Q=\\{\pm i,\pm j,\pm k,+1,-1\\},$ (20)

which is nothing but the non-Abelian quaternion group, the leading role in
non-Abelian band topology. The associated non-Abelian group multiplications of
the quaternion group are graphically represented in Fig. LABEL:figure2(a),
which shows that $i$, $j$ and $k$ satisfy $ij=k$, $jk=i$, $ki=j$ and
$i^{2}=j^{2}=k^{2}=-1$. The topological charges of band gaps or band nodes in
a $PT$-symmetric three-band spinless system can be characterized by these non-
Abelian group elements. Taking the $PT$-symmetric three-band insulator system
in one dimension as an example, the quaternion charges are defined by the
rotation of the 3-eigenstate frame when $k$ runs from $-\pi$ to $\pi$ across
the first Brillouin zone, as visualized in Figs.
LABEL:figure2(b)-LABEL:figure2(e). Specifically, the $\pi$ rotation of the
frame around the first, the second and the third eigenstates induces the non-
Abelian topological charges of $+i$, $+j$ and $+k$, respectively, for the
global three bands. The $2\pi$ rotation of the frame around any one of the
eigenstates constitutes the non-Abelian topological charge of $-1$. The
identity charge $1$ indicates no rotation and thus corresponds to the trivial
phase. The conjugate partners $-i$, $-j$ and $-k$ denote the inverse
rotations. Remarkably, the elements of $Q$ are observable only up to
conjugacy, which results from the sign freedom ±1 of real-valued eigenstates
[316]. The $PT$-symmetric three-band spinless system is the minimal model to
encode the non-Abelian band topology.

In the wake of Wu’s pioneering work [310], several theoretical [318, 319, 320,
321, 322, 323, 316, 324, 325, 326, 327, 328, 329, 330, 331] researches sprung
up and have witnessed the rise of the non-Abelian band topology. For instance,
the phonon spectra in layered silicates were predicted to host the non-Abelian
physics [323]. By far, thanks to the designability of artificial materials,
the non-Abelian band topology has been proposed, manipulated and probed in
various metamaterials such as transmission line networks [317, 332], phononic
crystals [333, 334, 335] full-dielectric [336] and biaxial photonic crystals
[337, 338, 332, 339, 340] and spring-mass systems [341]. Among them, the first
direct observation of the non-Abelian topological charges was achieved by Guo
et al. [317] in a one-dimensional $PT$-symmetric three-band insulator system
using transmission line network metamaterials. The different non-Abelian
quaternion charges, as represented in Figs. LABEL:figure2(b)-LABEL:figure2(e),
were extracted through the experimentally obtained eigenstate frame sphere.

Meanwhile, most researches about the non-Abelian band topology focus on three-
band semimetal systems in two or three dimensions with entangled band nodes
such as Weyl points, nodal lines, and nodal links. The possible configurations
of semimetals are set strict constraints by non-Abelian charges, which are
defined according to the rotation of the 3-eigenstate frame around a closed
loop in momentum space encircling the band nodes, as sketched in Fig.
LABEL:nodal(a). Different configurations offer an excellent platform to
explore the novel non-Abelian phenomena that are absent in one dimension, such
as the non-Abelian braiding of topological charges and their multipath phase
transitions. As shown in Fig. LABEL:nodal(b), braiding a band node with charge
$-i$ in the first gap around one with charge $-j$ in the second gap (III)
changes their charges by a factor of $-1$, which is the nontrivial
manifestation of the non-Abelian braiding, i.e., $(-j)(-i)(-j)^{-1}=i$ and
$(-i)(-j)(-i)^{-1}=j$. After braiding, the two band nodes in each gap have the
same charge (II). Such two nodes are topologically obstructed to annihilate,
as the total charge is $i^{2}=j^{2}-1$, which is the key signature of a
nontrivial braiding process. The non-zero integer-valued patch Euler class
$\xi$ integrated over a Brillouin zone patch enclosing the pair of band nodes,
defined as follows,

$\int_{k\in Brillouin\,zone\,patch}\langle{\nabla u_{1}(k)}|\times|\nabla
u_{2}(k)\rangle,$ (21)

can then quantifies their stability [333].

The braiding and phase transitions of non-Abelian charged band nodes were
first observed by Jiang et al. using tunable acoustic semimetals based on the
kagome model [333]. The kagome model with only the nearest and next-nearest
neighbor couplings [see Fig. LABEL:jiang(a)] results in a two-dimensional
$PT$-symmetric three-band semimetal. In experiments, the acoustic kagome
lattices were 3D printed using photosensitive resin [see Fig. LABEL:jiang(b)].
The sites of the kagome model are realized by cylindrical acoustic resonators.
The nearest and next-nearest neighbor couplings are realized by horizontal
tubes with different diameters that connect the resonators. Correspondingly,
the first three $s$-orbital bands of the acoustic kagome lattice are always
tangled together and the non-Abelian topological charges can be defined upon
these band nodes. By adjusting geometric parameters of metamaterials, in other
words, tuning the nearest and next-nearest neighbor couplings, the non-Abelian
node charges are converted through the braiding of nodes in adjacent gaps, as
well as their creation, annihilation, merging, transferring, and splitting. As
implied in the evolution of the experimentally measured band dispersions in
Fig. LABEL:jiang(c), these detailed dispersions are associated with different
non-Abelian topological charges. In each panel, the evolution tendency of node
charges is denoted by the arrows in the inset. The labels of different charges
are shown in Fig. LABEL:jiang(a) for reference. Specifically, when the next-
nearest neighbor couplings vanish [see the first panel of Fig.
LABEL:jiang(c)], Dirac points emerge at two $K$ points with charges $\pm i$ in
the first band gap and a quadratic node with charge $-1$ emerges at the
$\Gamma$ point. The quadratic node contributes to a nonzero patch Euler class
of $\xi=1$, which remains during the tuning process, proving the topological
robustness of the non-Abelian charge $-1$. Then, by continuously decreasing
and increasing the nearest and next-nearest neighbor coupling strength,
respectively, rich non-Abelian topological transitions occur. First, six pairs
of Dirac nodes with charges $\pm j$, of which the total charge is 1, are
created ”in a vacuum” on the $\Gamma K$ and $\Gamma M$ lines in the second
band gap. In the meantime, two halves of them head to $K$ and $M$ points,
respectively. Once the six Dirac nodes arrive at $K$ points, linear triply-
degenerate points emerge, with charges $\pm k$ since $(\pm i)(-j)=\mp k$ [see
the second panel in Fig. LABEL:jiang(c)]. Further tuning splits the triply-
degenerate points and leads to $\pm j$ charged Dirac nodes at $K$ points in
the first gap and $\pm i$ charged Dirac nodes at $KM$ lines in the second gap
[see the third panel in Fig. LABEL:jiang(c)]. The $\pm i$ charged Dirac nodes
continue to move and encounter the remaining six $-j$ charged Dirac nodes in
the second gap at three $M$ points, which results in the formation of $-1$
charged linear triply-degenerate points [see the fourth panel in Fig.
LABEL:jiang(c)]. Later, the linear triply-degenerate points split again,
giving rise to six $\pm i$ ($\pm j$) charged Dirac nodes at $\Gamma M$ ($KM$)
lines in the first (second) gap. Finally, the quadratic nodes at $K$ points
with charge $-1$ are created, forming the nonzero patch Euler classes of
$\xi=\pm 1$. Notably, the linear triply-degenerate points are the unavoidable
nodes for the braiding of two charges in adjacent gaps. Such triply-degenerate
points are widely studied before and after the birth of the non-Abelian band
topology [343, 344, 345, 346, 347, 348, 349, 350]. The multi-gap bulk-boundary
correspondences with both zigzag and armchair boundaries were also studied in
experiments, which provide an indirect manifestation of the braiding of the
band nodes, due to the fact that the Zak phase switches when passing through
an odd number of Dirac nodes.

Soon after, Wang et al. gave the first experimental observation of the earring
nodal links characterized by the non-Abelian quaternion charge of $-1$ [338].
They fabricated a three-dimensional $PT$-symmetric phononic crystal based on a
carefully designed $k\cdot p$ Hamiltonian. The phononic crystal is constructed
layer-by-layer along the $z$-direction with a unit cell stacked by two twisted
cuboids [see Fig. LABEL:jiangliu(a)]. The geometry has two mirror symmetries
and inversion symmetry which offers the opportunity that a nontrivial
accidental triple degenerate node can emerge within three bands above the
lowest band [see Fig. LABEL:jiangliu(b)]. The rotation of the 3-eigenstate
frame on the unit sphere along the closed loop encircling the node shows that
all three eigenstates rotate $2\pi$ around the $z$ axis [see Fig.
LABEL:jiangliu(c)], which denotes the key feature of the non-Abelian quaterion
charge $-1$. By small perturbation, the triple degenerate node can reduce to
two kinds of earring nodal links where their link cannot be fulley gapped,
which indicates the topological protection of quaterion charge $-1$. The
earring nodal links were verified experimentally in band dispersions at
different frequencies by performing three-dimensional Fourier transforms of
the scanned fields. Compared to Jiang’s work, Qiu et al. realized the similar
nontrivial non-Abelian braiding process of band nodes in acoustic
metamaterials based on a square lattice model [334]. However, taking a step
forward, they provided the first experimental evidence of the stability of
pair nodes of non-Abelian charges [see Fig. LABEL:jiangliu(d)]. Moreover, the
mirror eigenvalues were measured to characterize the braiding process.

The non-Abelian band topology is also generated to four-band $PT$-symmetric
systems [332], Floquet systems [327], non-Hermitian territory [326, 351, 352],
nontrivial Euler insulators [353, 324, 354, 335, 355] and so on. Notably, by
introducing a nonzero second next-neighbor couplings in the acoustic Kagome
lattice [see Fig. LABEL:jiangliu(e)], the so-called Euler insulator is formed
with the gapped lowest band [see Fig. LABEL:jiangliu(f)] captured by a meronic
Skymion texture [see Fig. LABEL:jiangliu(g)]. Such texture is based on the
orientation of real eigenstates in the two-dimensional Brillouin zone [335].
Recent work reveals that edge state distributions of higher-order topological
phases can be explained from a non-Abelian perspective [325]. We remark that a
further study on non-Abelian band topology in both theory and experiments is
deserved, as it is of fundamental importance using a global methodology to
understand the novel topological phenomena which cannot be characterized by
conventional band topology.

###  3.9 Topological defects and bulk-defect correspondences

The bulk-boundary correspondence, stating that the nontrivial bulk band
topology can be manifested as the existence of topologically protected lower-
dimensional boundary states, is at the core of the topological band theory.
Unprecedented progresses have been achieved in experimental verification of
band topology by virtue of probing spectral signals at boundaries including
surfaces, hinges or corners. Aside from boundaries which can be deemed as a
sort of crystal defects due to the lattice destruction, there is a variety of
defects that are embodied inside of crystals, such as vacancies,
substitutions, dislocations and disclinations. They substantially exist in
natural nanomaterials and nonequilibrium progresses [356, 357]. These defects
can be regarded as, to some extent, internal “boundaries” inside the bulk. A
question is naturally remarked whether the bulk band topology can be revealed
by these internal “boundaries”. One positive answer is provided by topological
defects [358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368], which
include, for example, dislocations [see Figs. LABEL:defect(a) and
LABEL:defect(b)] and disclinations [see Fig. LABEL:defect(c)].

Topological defects are robust lattice distortion against local lattice
deformation, distinguished from other defects by their stable topological
charges such as Burger’s vectors and Frank angles defined in real space [see
Fig. LABEL:defect]. They provide an exceptional methodology to explore the
topological band theory and extend the conventional bulk-boundary
correspondence to the bulk-defect correspondence. The interplay between the
real-space topological charges and the reciprocal-space band topology gives
rise to the underlying principles. Such dual topology in both real- and
reciprocal-space offers unprecedented opportunities in probing and inducing
topological phenomena such as topological defect states and fractional
spectral charges. Beside pieces of research in natural materials [369, 370],
most topological phenomena triggered by topological defects were demonstrated
in, e.g., photonic, phononic, and electrical circuit systems, owing to the
access to desirable lattice distortion in metamaterials. We here briefly
review the experimental progress of topological phenomena induced by
topological defects, particularly in phononic metamaterials. We proceed
according to the classification of topological defects.

An edge dislocation is characterized by the Burger’s vector, which acts as a
zero-dimensional defect in two dimensions. Topological states bound to edge
dislocations were earlier observed in mechanical metamaterials based on
deformed kagome lattices [371], as shown in Fig. LABEL:mechanical. The
topological bound states arise from a delicate interplay between the momentum
Berry phase and the Burger’s vertor in real space. Latter, topological zero-
dimensional light-trapping on an edge dislocation is directly measured in
photonic crystals by Li et al. [372], based on the topological edge states of
Chern insulators. More recently, topological dislocation states were
experimentally confirmed in a “weak” topological insulator constructed from
stacked one-dimensional Su-Schrieffer-Heeger chains within an anisotropic
magneto-mechanical system [373]. The first observation that partial edge
dislocations can probe the higher-order band topology in two-dimensional and
three-dimensional multipole insulators was implemented in circuit-based
resonator arrays [374]. Besides, by introducing additional two synthetic
translation dimensions, Chen et al. designed in two-dimensional dielectric
photonic crystals a four-dimensional topological insulator phase with a
nontrivial second Chern number [375]. The zero-dimensional dislocation states
in non-synthetic space evolve into chiral gapless one-dimensional dislocation
states in four-dimensional synthetic space due to the nontrivial second Chern
number therein, which were also validated in experiments, These topological
phenomena are yet to be explored in phononic systems.

As a matter of fact, topological defects first come to play in topological
materials since Ran et al. in 2009 made the first prediction that in weak
three-dimensional topological insulators [359], the screw dislocation line can
act the same role as boundaries to host one-dimensional topological gapless
helical boundary states. A screw dislocation characterized by the Burger’s
vector is a translation-breaking defect where the lattice planes follow
spirally along the dislocation line. The emergence of topological dislocation
states is jointly dictated, as follows, by the weak topological invariants
$v_{i}$ of occupied bands and the Burger’s vector $\bm{B}$ of the dislocation,

$\frac{1}{2}\,\bm{B}\cdot\sum_{i}v_{i}\bm{G}_{i}=\pi\,(mod\,2\pi),$ (22)

where $\bm{G}_{i}$ are the basis vectors of the reciprocal lattice. The
helical spin-polarized dislocation states were recently observed by Ye et al.
[376] and Xue et al. [377], respectively, in acoustic weak three-dimensional
topological insulators implemented by coupled resonators [see Figs.
LABEL:liulin(a)-LABEL:liulin(c) from Ye’s work]. In a different manner, the
former work is based on stacked layers, while the latter utilizes a
mathematical technology of pseudospin transformation [see Fig.
LABEL:liulin(d)]. More recently, a photonic realization of dislocation states
with a synthetic dimension has also been achieved in a coupled optical
waveguide array [379].

In the meantime, a unique mechanism of the screw dislocation termed
topological Wannier cycles that are irrelevant of weak topological invariants
were first proposed and experimentally demonstrated by Lin et al. [378], in an
acoustic higher-order topological insulator based on the stacked two-
dimensional Su-Schrieffer-Heeger model [see Fig. LABEL:liulin(e)]. In this
work, the screw dislocation plays the role of artificial local gauge flux
insertion [see Fig. LABEL:liulin(f)]. The disconnected four bands composed of
Wannier orbitals with different rotation symmetry representations evolve
cyclically into each other versus the local gauge flux, which is manifested as
spectral flows traveling multiple band gaps and topological boundary states
localized at the dislocation [see Fig. LABEL:liulin(g)]. Topological Wannier
cycles provide an alternative way in experiments to probe the higher-order
band topology when the spectral signals at corners are unavailable in the
absence of chiral symmetry.

Now let us turn to disclinations. Disclinations are zero- and one-dimensional
topological defects in two- and three-dimensions, respectively, which,
distinct from the breaking of translation symmetry of dislocations, disrupt
the rotation symmetry of the pristine lattice by “adding” or “subtracting” a
fan-shaped sector and then applying the cut-and-glue process [see Fig.
LABEL:defect(c)]. The real-space topology in disclinations is characterized by
the Frank angle. Topological responses induced by disclinations include