22footnotetext: The integer topological invariant is sufficient to
characterize braids that correspond to unknots, while additional topological
invariants are needed for a complete characterization of knotted braids.

# Non-Abelian Braiding of Topological Edge Bands

Yang Long1 These authors contributed equally to this work. Zihao Wang1 These
authors contributed equally to this work. Chen Zhang2 Haoran Xue3 Yuxin
Zhao4,5 Corresponding author<EMAIL_ADDRESS>Baile Zhang1,6 Corresponding
author<EMAIL_ADDRESS>1Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, Singapore 637371, Singapore
2 National Laboratory of Solid State Microstructures and Department of
Physics, Nanjing University, Nanjing 210093, China
3 Department of Physics, The Chinese University of Hong Kong, Shatin, Hong
Kong SAR, China
4 Department of Physics and HKU-UCAS Joint Institute for Theoretical and
Computational Physics at Hong Kong, The University of Hong Kong, Hong Kong,
China
5 HK Institute of Quantum Science& Technology, The University of Hong Kong,
Hong Kong, China
6 Centre for Disruptive Photonic Technologies, Nanyang Technological
University, Singapore 637371, Singapore

###### Abstract

Braiding is a geometric concept that manifests itself in a variety of
scientific contexts from biology to physics, and has been employed to classify
bulk band topology in topological materials. Topological edge states can also
form braiding structures, as demonstrated recently in a type of topological
insulators known as Möbius insulators, whose topological edge states form two
braided bands exhibiting a Möbius twist. While the formation of Möbius twist
is inspiring, it belongs to the simple Abelian braid group $\mathbb{B}_{2}$.
The most fascinating features about topological braids rely on the non-
Abelianness in the higher-order braid group $\mathbb{B}_{N}$ ($N\geq 3$),
which necessitates multiple edge bands, but so far it has not been discussed.
Here, based on the gauge enriched symmetry, we develop a scheme to realize
non-Abelian braiding of multiple topological edge bands. We propose tight-
binding models of topological insulators that are able to generate topological
edge states forming non-Abelian braiding structures. Experimental
demonstrations are conducted in two acoustic crystals, which carry three and
four braided acoustic edge bands, respectively. The observed braiding
structure can correspond to the topological winding in the complex eigenvalue
space of projective translation operator, akin to the previously established
point-gap winding topology in the bulk of the Hatano-Nelson model. Our work
also constitutes the realization of non-Abelian braiding topology on an actual
crystal platform, but not based on the “virtual” synthetic dimensions.

††preprint: AIP/123-QED

Braiding is a multifaceted concept that finds applications across diverse
scientific domains, from physics to biology [1, 2, 3]. Geometrically described
by the intricate intertwining of strands forming specific patterns, braiding
has significantly broadened its scope in physical systems [4, 5, 6, 7, 8, 9],
including the classification of topological bulk bands [10, 11, 12].
Topological edge states can also form braiding structures. For example, in the
recently demonstrated Möbius topological insulators [13, 14, 15, 16, 17, 18] ,
the topological edge states can form two braided edge bands exhibiting a
Möbius twist. The time-reversal-invariant Möbius insulator phase is protected
by the projective translation symmetry and is characterized by the
$\mathbb{Z}_{2}$ topological invariant [15, 16, 17].

While the presence of Möbius twist enriches the topological classification, it
is worth noting that the Möbius twist represents an Abelian braid formed by
two bands, thus falling within the Abelian braid group $\mathbb{B}_{2}$. This
prompts a fundamental question: will there be topological insulators beyond
the Möbius topological insulators, namely, the topological insulators whose
edge states can form non-Abelian braids of multiple edge bands? Note that the
non-Abelian braiding topology has sparked some recent experimental progresses
in either synthetic dimensions [19, 8] or electric circuit networks [20, 21],
but has not been realized in an actual crystal platform, neither has been
applied to topological edge bands. In this work, we propose topological
insulators protected by translation symmetry, mirror symmetry and time-
reversal symmetry, which can realize non-Abelian braiding of topological edge
bands.

Conventionally a translation operator translating a crystal by $\bm{a}$ is
represented as $e^{i\bm{k}\cdot\bm{a}}$, which exerts no constraint on the
$k$-space Hamiltonian. Therefore, translation symmetry is usually not
considered as protecting symmetry of topological insulators. However, recent
research has shown that this common belief no longer holds when the gauge
structure of crystals is considered [15, 22, 23, 24]. Particularly, for
acoustic crystals gauge structures ubiquitously exist and can be flexibly
engineered [16, 17, 18, 25, 26, 27, 28, 29], and therefore it is important to
consider symmetry constraints from translation symmetry for topological
acoustic crystals [30, 31, 32, 33, 34].

Let us briefly introduce the underlying reason. The gauge connection
configuration of a non-interacting tight-binding model consists of the phases
of hopping amplitudes [see Fig. 1(a) for an example]. Under a given
configuration $\mathcal{C}$, we can choose primitive unit cells and the
corresponding translation symmetry operators $\hat{\Gamma}_{i}$ with
translation vectors $\bm{a}_{i}$. Here, the gauge connection configuration
$\mathcal{C}$ is invariant under $\hat{\Gamma}_{i}$. Nevertheless,
$\hat{\Gamma}_{i}$ may not be the minimal translations considering the gauge
structure. There may exist a translation operator $\hat{L}_{i}$ with
fractional translation vector $\bm{a}_{i}/N$ for some $i$, such that
$\hat{L}_{i}$ transforms $\mathcal{C}$ to $\mathcal{C}^{\prime}$ [see Fig.
1(b)], where $\mathcal{C}^{\prime}$ is related to $\mathcal{C}$ by a gauge
transformation $\mathcal{G}_{i}$ [see Fig. 1(c)]. Then, the combination
$\hat{\mathcal{L}}_{i}=\mathcal{G}_{i}\hat{L}_{i}$, termed
$\mathcal{G}$-dressed translation, is a symmetry of the system [as
demonstrated in Fig. 1(a), (b) and (c) as a cycle], namely
$[H,\hat{\mathcal{L}}_{i}]=0$.

Figure 1: Illustration for $\mathcal{G}$-dressed translation symmetry. (a) A
1D chain with a gauge connection configuration $\mathcal{C}$. All hopping
amplitudes have real values with the same magnitude. Negative and positive
ones are marked in red and blue, respectively. Accordingly, $\mathcal{C}$ is
visualized as the blue-red pattern. $\hat{\Gamma}$ is the unit translation of
the primitive unit cells with one surrounded by the dashed line. (b) The
translation of the 1D chain by $\hat{L}$. $\hat{L}$ corresponds to a half of
$\hat{\Gamma}$ as indicated in (a). $\mathcal{C}$ in (a) is changed to
$\mathcal{C}^{\prime}$ in (b). (c) The gauge transformation $\mathcal{G}$ on
the 1D chain in (b). $\mathcal{G}$ is represented as $\pm$ sign on each site.
The gauge transformation switches the color of all bonds as each bond has
different signs at its two endpoints. Hence, (b) is transformed to (a) by
$\mathcal{G}$.

We choose the $\mathcal{G}$-dressed translation symmetry
$\hat{\mathcal{L}}_{x}$ as a protecting symmetry of the topological insulators
together with mirror and time-reversal symmetries. In each eigenspace of
$\hat{\mathcal{L}}_{x}$, the topological invariant leads to an edge bands on
the edges preserving $\hat{\mathcal{L}}_{x}$. In our work, there are totally
$N$ edge bands and each edge band is charged by an eigenvalue of
$\hat{\mathcal{L}}_{x}(k_{x})$. The $N$ edge bands can be regarded as strands
of a braid in the (${\rm Re}[\mathcal{L}_{x}]$, ${\rm Im}[\mathcal{L}_{x}]$,
$k_{x}$)-space (${\rm Re/Im}[\cdot]$ denotes the real/imaginary part), which
will be denoted as the $\mathcal{L}_{x}$-$k_{x}$ space hereafter. In the
$\mathcal{L}_{x}$-$k_{x}$ space, the topological edge states can form a non-
Abelian braiding structure.

Figure 2: Twisted bandstructures of multiple topological edge bands in the
$E$-$k$ space and their braiding in the complex $\mathcal{L}_{x}$-$k_{x}$
space.  (a) In the Möbius topological insulator, two initially separated edge
bands can braid with each other after traversing through the edge BZ, forming
a Möbius twist. The $\psi_{s}$ denotes the edge eigenstate. Since the momentum
$k_{x}$ is restricted to the edge BZ $k_{x}\in[-\pi,\pi]$, each topological
edge state $\psi_{s}$ can be represented as a strand of a braid in the
$\mathcal{L}_{x}$-$k_{x}$ space. (b) The Möbius twist corresponds to an
Abelian braid $\tau_{1}$ in the $\mathcal{L}_{x}$-$k_{x}$ space. By exploiting
the projective translation symmetry, we can realize the twisted bandstructures
of multiple edge bands, such as three edge bands in (c) and four edge bands in
(e). These twisted bandstructures correspond non-Abelian braids in the
$\mathcal{L}_{x}$-$k_{x}$ space, which are described by
$\tau_{1}^{-1}\tau_{2}^{-1}$ in (d) and
$\tau_{2}^{-1}\tau_{1}^{-1}\tau_{3}^{-1}$ in (f), respectively. (g,h) Non-
Abelian properties of $\tau_{1}^{-1}\tau_{2}^{-1}$ and
$\tau_{2}^{-1}\tau_{1}^{-1}\tau_{3}^{-1}$.  Figure 3: Topological insulators
with braided edge bands. (a) The unit cell of the topological insulator with
three braided edge bands. Here, ($n,m$) ($n,m\in\mathbb{Z}$) denotes the
coordinate of the site. The dotted gray box denotes the primitive unit cell.
(b) The unit cell of the topological insulator with four braided edge bands.
Here, $\kappa_{1}=-\kappa_{2}$ due to the projective translation symmetry,
where $\kappa_{1}$, $\kappa_{2}\in\mathbb{R}$ and $t_{1}$, $t_{2}>0$. (c) The
bulk band of the topological insulator in (a) when $t_{1}\neq t_{2}$,
$\kappa_{1}=-\kappa_{2}=-0.1$. The system will open a gap and become
topological (trivial) when $t_{1}=0.2$ and $t_{2}=1.0$ ($t_{1}=1.0$ and
$t_{2}=0.2$). (d) The bulk band of the topological insulator in (b) when
$t_{1}\neq t_{2}$, $\kappa_{1}=-\kappa_{2}=-0.1$. The system will open a gap
and become topological (trivial) when $t_{1}=0.2$ and $t_{2}=1.0$ ($t_{1}=1.0$
and $t_{2}=0.2$). The right two figures in (c,d) show the edge spectra, and
the gray regions in them denote the bulk bands.

We begin by introducing the twisted energy bandstructures. As depicted in Fig.
2(a), recent studies in the Möbius topological insulator [17, 16, 15, 17] have
shown that, under projective translation symmetry, two initially separated
edge bands can braid with each other after traversing through the edge BZ,
forming a Möbius twist. The topology of the Möbius twist can be characterized
by an Abelian braiding $\tau_{1}^{-1}$ between two bands in the
$\mathcal{L}_{x}$-$k_{x}$ space in Fig. 2(b), where $\tau_{n}$
($\tau_{n}^{-1}$) is the braid operation with the meaning that the $n$-th
strand crosses under (above) the ($n+1$)-th strand. In contrast, our 2D
topological insulators can accommodate multiple topological edge states and
braid them together, with three bands illustrated in Fig. 2(c) and four bands
illustrated in Fig. 2(e), respectively. Now each edge state can return to its
original status after traversing through the edge BZ for three or four times.
These three or four topological edge bands then form non-Abelian braids:
$\tau_{1}^{-1}\tau_{2}^{-1}$ in Fig. 2(d) and
$\tau_{2}^{-1}\tau_{1}^{-1}\tau_{3}^{-1}$ in Fig. 2(f), respectively. Namely,
the braids cannot be topologically equivalent after exchanging braid
operations, as shown in Fig. 2(g,h).

To realize the topological edge states depicted in Fig. 2(c,e), we propose the
tight-binding models shown in Fig. 3(a,b). The Hamiltonians for these models
preserve the projective translation symmetry $\hat{\mathcal{L}}_{x}$, mirror
symmetry $\hat{\mathcal{M}}_{y}$ ($y\rightarrow-y$), and time-reversal
symmetry $\hat{\mathcal{T}}$. The gauge transformations to be combined with
the translation operators $\hat{L}_{x}$ are given by:
$\mathcal{G}=\mathbb{I}_{2}\otimes{\rm diag}[1,1,-1]$ in Fig. 3(a) and
$\mathcal{G}=\mathbb{I}_{2}\otimes{\rm diag}[1,1,1,-1]$ in Fig. 3(b). The
gauge transformation $\mathcal{G}$ is denoted as the sign $\pm$ on the sites.
The introductions of the gauge transformations lead to $3\times 2$ sites and
$4\times 2$ sites for the primitive unit cells, respectively. As a result of
projective translation symmetry, we have $\kappa_{2}=-\kappa_{1}$ in both
cases. For the lattice in Fig. 3(a), we can introduce a unitary transformation
$U$ to diagonalize $\hat{\mathcal{L}}_{x}$:
$U\hat{\mathcal{L}}_{x}U^{\dagger}=e^{i\frac{k_{x}}{3}}{\rm
diag}[-1,e^{i\pi/3},e^{-i\pi/3}]\otimes\mathbb{I}_{2}$. Since
$[\hat{\mathcal{L}}_{x},H]=0$, the Hamiltonian $H$ can be represented in a
block diagonalized form $\mathcal{H}=UHU^{\dagger}={\rm
diag}[h_{1},h_{2},h_{3}]$. Because $\hat{\mathcal{L}}_{x}^{3}=-e^{ik_{x}}$,
$U$ is not periodic when $k_{x}$ goes from $-\pi$ to $\pi$ (i.e.
$U|_{k_{x}=-\pi}\neq U|_{k_{x}=\pi}$), and $\mathcal{H}$ is also not periodic
along $k_{x}$ (i.e. $\mathcal{H}|_{k_{x}=-\pi}\neq\mathcal{H}|_{k_{x}=\pi}$).
We find that the blocks in $\mathcal{H}$ have the following relation 111 See
Supplemental Material at [link] for more details about theoretical analysis
about Hamiltonian and topological properties, non-trivial knots or links,
experimental discussions, and discussions about the braiding topology in the
$\mathcal{L}_{x}$-$k_{x}$ space, which include the Ref. [52, 53] :

$h_{3,2,1}(k_{x}+2\pi,k_{y})=h_{2,1,3}(k_{x},k_{y}).$ (1)

Here Eq. 1 implies that the blocks can transform into each other in a cyclic
manner when $k_{x}$ goes from $-\pi$ to $\pi$. This gives rise to the 3-fold
rotation $\hat{C}_{3}$ behavior in the basis $\\{h_{3},h_{2},h_{1}\\}$[35].
Similarly, the lattice in Fig.3(b) satisfies
$h_{4,3,2,1}(k_{x}+2\pi,k_{y})=h_{3,2,1,4}(k_{x},k_{y})$, leading to the
4-fold rotation $\hat{C}_{4}$ behavior in the basis of blocks in $\mathcal{H}$
[35].

Figure 4: Experimental measurements of twisted edge bandstructures in
topological acoustic lattices. (a) A snapshot of the acoustic crystal
corresponding to the tight-binding model in Fig. 3(a). The unit cell is
denoted in a dashed rectangle. Here, $a_{x}=180$mm, $a_{y}=120$mm. (b) A
snapshot of the acoustic crystal corresponding to the tight-binding model in
Fig. 3(b). Here, $a_{x}=200$mm, $a_{y}=100$mm. (c) Simulated bulk
bandstructure of the lattice in (a). The blue region denote the bulk band gap.
(d) Simulated bulk bandstructure of the lattice in (b). (e) Experimentally
measured edge spectrum of the acoustic lattice in (a). The background color
denotes the Fourier spectrum of the measured pressure field distribution data,
and the dotted white lines represent the theoretical predictions with fitted
parameters. (f) Experimentally measured edge spectrum of the acoustic lattice
in (b). Here, we use $k_{x,y}$ to denote $k_{x,y}a_{x,y}$, $\psi_{n}$ denotes
the $n$-th edge state.  Figure 5: Experimental measurements of non-Abelian
braiding of topological edge bands in the $\mathcal{L}_{x}$-$k_{x}$ space. (a)
Three topological edge bands form a non-Abelian braid
$\tau_{1}^{-1}\tau_{2}^{-1}$. The lines and spheres denote the theoretical
predictions and calculated values based on experimental measurements,
respectively. The colors of lines correspond to the edge states in Fig. 4(e).
(b) The knot obtained by presenting the braid in (a) as a torus knot. The gray
line represent the theoretical prediction, while the calculated values based
on experimental measurements are denoted by the spheres. (c) The winding of
$\mathcal{L}_{x}$ in (a) on the (${\rm Re}[\mathcal{L}_{x}]$, ${\rm
Im}[\mathcal{L}_{x}]$) plane. The positions of points denote the different
values of $\mathcal{L}_{x}$ , with the color representing the wave vector
$k_{x}\in[-\pi,\pi]$. The winding behaviour can be represented by a winding
number $v=1$. (d) Four topological edge bands form a non-Abelian braid
$\tau_{2}^{-1}\tau_{1}^{-1}\tau_{3}^{-1}$. The colors of lines correspond to
the edge states in Fig. 4(f). (e) The knot structure obtained by presenting
the braid in (d) as a torus knot. (f) The winding of $\mathcal{L}_{x}$ in (d)
on the (${\rm Re}[\mathcal{L}_{x}]$, ${\rm Im}[\mathcal{L}_{x}]$) plane. The
winding behaviour can be represented by a winding number $v=1$.

With preserved $\hat{\mathcal{L}}_{x}$, $\hat{\mathcal{M}}_{y}$ and
$\hat{\mathcal{T}}$, the topological properties of lattices in Fig. 3(a,b) can
be described by a $\mathbb{Z}$ topological invariant [36, 37] (i.e.,
$M\mathbb{Z}$ topology in class AI), which can have topological edge states on
an edge protected by $\hat{\mathcal{L}}_{x}$, similar to the Möbius insulator
[15, 17, 16]. Because the preserved $\hat{\mathcal{L}}_{x}$ can block
diagonalize $H$ into $\mathcal{H}$ and the blocks in $\mathcal{H}$ can be
transformed from each other by the relation of Eq. 1, the $\mathbb{Z}$
invariant can be defined according to $h_{1}$. The $M\mathbb{Z}$ topology of
$h_{1}$ can be described by an integer number
$N_{-}=N_{-}^{\pi}-N_{-}^{0}\in\mathbb{Z}$ [37, 35], where
$N_{-}^{0}$($N_{-}^{\pi}$) denotes the number of occupied bands at
$k_{y}=0$($k_{y}=\pi$) with $-1$ eigvenvalues of $\hat{\mathcal{M}}_{y}$.
$N_{-}$ can associate with the Zak phase of 1D subsystem $h_{1}$ when $k_{x}$
is fixed [35]. Note that $N_{-}^{\pi}$ is constant for any $k_{x}$ because the
gap keeps open in the whole BZ. When $t_{2}>t_{1}$ ($t_{1}<t_{2}$), the
lattices in Fig.3(a,b) can open a bulk gap in Fig.3(c,d), respectively, and
become topological with $N_{-}=-1$ (trivial with $N_{-}=0$). In the
topological phase, non-trivial bulk topological invariant ($N_{-}=-1$) of each
1D subsystem $h_{i}$ can correspond to an in-gap topological edge state on the
edge that preserves $\hat{\mathcal{L}}_{x}$. As shown in Fig.3(c,d), multiple
in-gap edge states from subsystems $\\{h_{i}\\}$ can form the twisted
bandstructures, corresponding to interconnected bulk states (e.g., Eq. 1).
Because each subsystem $h_{i}$ belongs to the block with a distinct eigenvalue
of $\hat{\mathcal{L}}_{x}$, these edge states can thus form a non-Abelian
braid in the $\mathcal{L}_{x}$-$k_{x}$ space (we will discuss in the
following).

According to the tight-binding models in Fig. 3(a,b), we design the
topological acoustic crystals in Fig. 4(a,b), which exploit coupled acoustic
resonators that host dipolar resonances. The positive/negative couplings are
implemented by engineering the coupling channels [34]. The cuboid resonators
used in experiment have dimensions of $48\text{ mm}\times 24\text{ mm}\times
12\text{ mm}$ in Fig.4(a) and $40\text{ mm}\times 20\text{ mm}\times 10\text{
mm}$ in Fig.4(b). The resonators are hollow (filled with air) and surrounded
by hard walls. The geometrical details can be found in the Supplementary
Materials [35]. After obtaining the bandstructures of the acoustic crystals in
Fig.4(a,b) numerically, we observe gaps between $3.45\sim 3.7$ kHz and
$4.16\sim 4.4$ kHz, respectively, as shown in Fig.4(c,d). We experimentally
measure the acoustic pressures of each site on the edge in a finite sample and
obtain the edge bands after performing Fourier transformation, as shown in
Fig.4(e,f). The experimental results show the presence of topological edge
states in the gaps of bulk bands. Importantly, these edge states have twisted
bandstructures as predicted theoretically in Fig.3(c,d).

In the following, we will discuss the braiding topology of the topological
edge bands. While the single bulk topological invariant derived from bulk band
topology is effective in predicting the presence of topological edge states in
two different crsytals in Fig. 3(a,b), the edge bands can form different
braids. Since $[\hat{\mathcal{L}}_{x},H]=0$, namely $\hat{\mathcal{L}}_{x}$
shares the same eigenstates as $H$, we can investigate the braiding based on
$\hat{\mathcal{L}}_{x}$. To illustrate the braiding structure in Fig.4(e), we
numerically simulated the values
$\mathcal{L}_{x}=\langle\psi_{s}|\hat{\mathcal{L}}_{x}|\psi_{s}\rangle$ of
each edge state $|\psi_{s}\rangle$, and also calculated their values based on
the measured eigenstates $|\psi_{s}\rangle$ [35], as presented in Fig.5(a).
The complex values of $\mathcal{L}_{x}$ for three edge states in Fig.4(e) form
a braiding structure in the $\mathcal{L}_{x}$-$k_{x}$ space, which is a non-
Abelian braid described by the braid word $\tau_{1}^{-1}\tau_{2}^{-1}$
($\tau_{1}^{-1}\tau_{2}^{-1}\neq\tau_{2}^{-1}\tau_{1}^{-1}$) [35]. Despite
these three edge states crossing each other in the $E$-$k$ space, they will
satisfy the separable band condition [38] in the $\mathcal{L}_{x}$-$k_{x}$
space. Similarly, in Fig.5(d), the $\mathcal{L}_{x}$ values of edge states of
the lattice in Fig. 4(b) are simulated numerically and calculated from
measured data, forming a non-Abelian braid described by
$\tau_{2}^{-1}\tau_{1}^{-1}\tau_{3}^{-1}$ [35].

For three edge bands in Fig. 5(a), $N=3$, the topology of 3 such separable
bands is classified by the braid group $\mathbb{B}_{3}$. $\mathbb{B}_{3}$ is
isomorphic to the knot group of the trefoil knot $K$:
$\pi_{1}(\mathbb{R}^{3}\setminus K)$, which is an infinite non-Abelian group.
As the momenta at the two ends of the edge BZ are equivalent, the two ends of
the braid are identical, thus closing the braid to form a knot or a link.
Since all bands are connected, the braiding shall form a knot, as shown in
Fig. 5(b). The knot topology in Fig. 5(b) corresponds to an unknot. Because
the unknot has the knot group isomorphic to the group of integers
$\mathbb{Z}$, we can exploit an integer topological invariant to describe the
topological properties in the $\mathcal{L}_{x}$-$k_{x}$ space [10, 39]:
$v=\frac{1}{2\pi
i}\int_{0}^{2\pi}\partial_{k_{x}}\ln\det[\mathfrak{L}_{x}-\frac{1}{2}{\rm
Tr}(\mathfrak{L}_{x})]dk_{x}$, where $\mathfrak{L}_{x}$ is a matrix with the
element as
$(\mathfrak{L}_{x})_{m,n}=\langle\psi_{m}|\hat{\mathcal{L}}_{x}|\psi_{n}\rangle$,
and $|\psi_{n}\rangle$ denotes the $n$-th edge state. $v$ describes how many
times the bands braid as $k_{x}$ varies from $-\pi$ to $\pi$, with the sign
indicating the handedness of the braid, similar to the vorticity of separable
bands [38]. In Fig. 5(c), we find that $v=1$ for three edge bands in Fig.
5(a). Similarly, for four edge bands ($N=4$), it has an unknot in Fig. 5(e)
and belongs to the $\mathbb{Z}$ group, which has $v=1$, as shown in Fig. 5(f).

The braiding structures in Fig. 5 are protected by projective translation
symmetry $\hat{\mathcal{L}}_{x}$ and cannot be removed without breaking
$\hat{\mathcal{L}}_{x}$ or closing the band gap [35]. The meaning of $v$ can
be understood by mapping to the 1D Hatano-Nelson model within a superlattice
[35, 40, 41, 42, 43, 44]. For a large finite system truncated by open boundary
conditions, if the edge bands have that $v=1$, the $\hat{\mathcal{L}}_{x}$
spectrum of such a finite lattice forms a branch on the (${\rm
Re}[\mathcal{L}_{x}]$, ${\rm Im}[\mathcal{L}_{x}]$) plane, which is consistent
with the previous research works [35, 45, 10]. The non-trivial knots or links
of topological edge bands can be realized by introducing long-range couplings
[35].

###### Acknowledgements.

In summary, we demonstrate the non-Abelian braiding topology of topological
edge states and experimentally observe braiding structures in a topological
acoustic crystal. Our work opens up a new avenue for exploring braiding
topology and can be applied to understand braiding-related topological
phenomena [46, 47], such as braids in the isofrequency plane [48],
nonequilibrium stochastic dynamics [49] or non-Abelian photonics [50, 51].
This research is supported by Singapore National Research Foundation
Competitive Research Program Grant No. NRF-CRP23-2019-0007, Singapore Ministry
of Education Academic Research Fund Tier 2 Grant No. MOE2019-T2-2-085. H.X.
acknowledges support from the startup fund of The Chinese University of Hong
Kong. Y.L. gratefully acknowledges the support of the Eric and Wendy Schmidt
AI in Science Postdoctoral Fellowship, a Schmidt Futures program.

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