# Observation of non-Abelian topological semimetals and their phase
transitions

Bin Jiang 1∗, Adrien Bouhon2,∗,†, Zhi-Kang Lin1,∗,
Xiaoxi Zhou1, Bo Hou1, Feng Li1,5, Robert-Jan Slager3,⋆†, Jian-Hua Jiang1,6⋆†
1School of Physical Science and Technology and
Collaborative Innovation Center of Suzhou Nano Science and Technology,
Soochow University, 1 Shizi Street, Suzhou 215006, China
2Nordic Institute for Theoretical Physics (NORDITA),
Stockholm, Sweden
3TCM Group, Cavendish Laboratory, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
5 School of Physics, Beijing Institute of Technology, Beijing 100081, China
6 Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province
$\&$ Key Lab of Modern Optical Technologies of Ministry of Education, Soochow
University, Suzhou 215006, China.
†To whom correspondence should be addressed:
<EMAIL_ADDRESS>(AB<EMAIL_ADDRESS>(RJS) and<EMAIL_ADDRESS>(JHJ).
$\ast,\star$ These authors contributed equally

###### Abstract

Topological phases of matter lie at the heart of physics, connecting elegant
mathematical principles to real materials that are believed to shape future
electronic and quantum computing technologies. To date, studies in this
discipline have almost exclusively been restricted to single-gap band topology
because of the Fermi-Dirac filling effect. Here, we theoretically analyze and
experimentally confirm a novel class of multi-gap topological phases, which we
will refer to as non-Abelian topological semimetals, on kagome geometries.
These unprecedented forms of matter depend on the notion of Euler class and
frame charges which arise due to non-Abelian charge conversion processes when
band nodes of different gaps are braided along each other in momentum space.
We identify such exotic phenomena in acoustic metamaterials and uncover a rich
topological phase diagram induced by the creation, braiding and recombination
of band nodes. Using pump-probe measurements, we verify the non-Abelian charge
conversion processes where topological charges of nodes are transferred from
one gap to another. Moreover, in such processes, we discover symmetry-enforced
intermediate phases featuring triply-degenerate band nodes with unique
dispersions that are directly linked to the multi-gap topological invariants.
Furthermore, we confirm that edge states can faithfully characterize the
multi-gap topological phase diagram. Our study unveils a new regime of
topological phases where multi-gap topology and non-Abelian charges of band
nodes play a crucial role in understanding semimetals with inter-connected
multiple bands.

With the advent of topological phases, the past few years have seen rapid
progresses in classifying topological states of matter. Upon analyzing the
manner in which bands transform at high-symmetry momenta, one can deduce
compatibility relations Kruthoff  _et al._ (2017) conveying how many different
classes of gapped band structures exist on a lattice geometry with a given set
of symmetries (see Fig. 1A for schematic illustration). Moreover, by comparing
to real space, efficient schemes were proposed that can discern which of these
classes can be deemed topological by absence of an atomic limit Po  _et al._
(2017); Bradlyn  _et al._ (2017).

Here, we consider an exotic type of topological phases beyond the above
paradigms that, instead, depend on topological charge conversion processes
when band nodes are braided with respect to each other in momentum space or
recombined over the Brillouin zone. The braiding of band nodes is in some
sense the reciprocal space analog of the non-Abelian braiding of particles in
real space and is manifested by a new topological invariant, the Euler class,
$\xi$ Wu  _et al._ (2019); Ahn  _et al._ (2019); Bouhon  _et al._ (2020a).
Considering a kagome model and its realization in acoustic metamaterials, we
uncover a rich interplay between these concepts and crystalline symmetries Fu
(2011); Slager  _et al._ (2012); Bouhon and Black-Schaffer (2017); Fang  _et
al._ (2012); Slager (2019); Cornfeld and Carmeli (2021) and further
experimentally confirm the above physics in acoustic metamaterials by tuning
their geometry. The rich phase diagram features unavoidable intermediate
phases featuring band degeneracies at high symmetry point with unconventional
dispersion features that have a topological origin that is not captured by
representation theory alone Yang and Nagaosa (2014); Bradlyn  _et al._ (2016).
In addition, we reveal the edge physics associated with the non-Abelian band
braiding within the above context.

[h]

Figure 1: Conventional topological insulators (TIs) versus multi-gap topology
of Euler class. A, In the usual TI setting, the spectrum is divided in a
conduction and valence band sector (red and blue, respectively). Starting from
a trivial insulator (I), a TI (III) can be obtained from a band inversion
(II), which changes the character of the band when going from one high-
symmetry momentum $\Gamma_{0}$ to another $\Gamma_{1}$. This change is
schematically denoted as $\pm$, although this can be a general property of the
irreducible band representation at the high-symmetry momenta. B, In the Euler
case, one considers a refined partitioning of bands, for example a subblock
that has two band gaps. Starting from a trivial gapped system, band nodes can
be induced in both gaps (I). These nodes are oppositely charged (open/closed
symbols) within a gap and behave as quaternions $\pm i$ and $\pm j$ with
respect to the charges in the other gap (red circles vs. blue triangles). As a
result, braiding a band node of the first gap around one in the second gap
(III) along the brown trajectory changes its charge by a factor of $-1$,
resulting in a spectrum exhibiting nodes with similarly-valued frame charges
(II). The Euler class $\xi$ integrated over Brillouin zone patches (green
boxes) then quantifies the stability of the nodes (III and IV). C, (I) A Dirac
cone with a Berry phase $\gamma_{B}=\pi$. (II) A quadratic band touching point
stabilized by the patch Euler class $\xi=\pm 1$. (III) A linearly-dispersed
triply-degenerate point with a topological frame charge $q=\pm k$. (IV) A
parabolically-dispersed triply-degenerate point with a topological frame
charge $q=-1$. The latter can intuitively understood as the combination of a
double $i,j$ charge with a third band.

Euler class and multi-gap topology
While the study of topological materials and their interplay has resulted in a
thriving field on both the theoretical and experimental front, Euler class
provides a new paradigm of topological phases. Akin to Chern numbers, systems
that feature combined twofold rotation and time reversal symmetry,
$C_{2}\cal{T}$, or combined parity and time reversal symmetry, $\cal{PT}$, can
be described by a real Hamiltonian that can exhibit this topological invariant
Wu  _et al._ (2019); Ahn  _et al._ (2019); Bouhon  _et al._ (2020a). From a
more physical perspective, these new forms of topology can be best understood
as being the consequence of a perspective that takes into account multiple
band gap conditions Bouhon  _et al._ (2020b). For instance, one can revert to
a minimal Euler model that arises in a three-band system featuring two gapless
bands and a gapped band, as detailed in Fig. 1. The Euler invariant manifests
itself by the presence of a number of $2\xi$ irremovable band nodes between
the two gapless bands. Conventionally, semi-metals host nodes with opposite
topological charges that can be pairwise annihilated as the consequence of the
celebrated Nielsen-Ninomiya theorem. In the Euler case, however, the nodes
have an additional frame charge that is the same for the irremovable nodes.
The according inability to annihilate the band nodes reveals the topological
obstruction that the topological Euler number $\xi$ imposes.

The above counter-intuitive possibility is intimately tied to both the real-
valued form of the Hamiltonian and the concept of orientability. Namely, the
reality condition ensures that the set of Eigenstates can be interpreted as an
oriented orthonormal frame, or dreibein when restricting to three-band systems
as in this work. Decomposing the 3D rotation matrix along paths encircling
stable nodes we can then directly compute the acquired angle of the monopole,
or frame charge, reflecting the geometry of SO(3) rotations Johansson and
Sjöqvist (2004); Bouhon  _et al._ (2020a). Then, as a component of the
spectral decomposition of the Hamiltonian, this frame is only defined up to a
sign for each Eigenstate. Hence, the frame charges of the nodes are related to
the flag variety $\mathsf{SO}(3)/D_{2}$ Wu  _et al._ (2019); Bouhon  _et al._
(2020a). As a consequence, the band nodes correspond to $\pi$-disclinations in
a biaxial nematic liquid crystals Alexander  _et al._ (2012); Liu  _et al._
(2016); Volovik and Mineev (2018) and the charges $q$ associated to nodal
points within three bands take values in the conjugacy classes $\pm q$ of the
quaternion group, i.e. with $q\in\\{1,i,j,k,-1\\}$. Specifically, in our case
$\pm i$ and $\pm j$ characterize single nodes in each of the respective gaps,
$\pm k$ represents a pair of nodes in each gap, and $-1$($=i^{2}=j^{2}$)
captures the stability of two nodes of the same gap Wu  _et al._ (2019);
Tiwari and Bzdušek (2020).

Using the fact that frame charges behave as quaternions, we then directly
uncover a geometric relation between the braiding process involving the two
gaps Wu  _et al._ (2019); Ahn  _et al._ (2019); Bouhon  _et al._ (2020a), as
explained in Fig. 1B, and the Euler invariant. Specifically, one can start
from a trivially gapped system and create pairs of band nodes in the two gaps
having charges $\pm i$ and $\pm j$, respectively. Each gap hosts a pair of
oppositely-valued band node charges, which may be gapped out in a reverse
process to arrive back at the trivial system (i.e., Euler class $\xi=0$).
However, braiding a node residing in the first band gap, say $-i$, along a
node in the second gap, say $-j$, produces in each gap a pair of nodes that
cannot be annihilated upon collision (non-zero Euler class), which can
effectively be phrased in terms of the conversion of the non-Abelian charges
of the braided nodes rendering, e.g., the combination $\\{+i,+i,+j,+j\\}$ Wu
_et al._ (2019); Ahn  _et al._ (2019); Bouhon  _et al._ (2020a), see Fig. 1.
We note that an additional trivial band can thus trivialize the Euler model by
the reverse braid process, showing that this enigmatic topology is formally a
form a fragile topology, albeit rather distinct from the symmetry eigenvalue
indicated variant Po  _et al._ (2018); Bouhon  _et al._ (2019); Song  _et al._
(2019); Peri  _et al._ (2020); Song  _et al._ (2020).

In the above we have implicitly applied the concept of patch Euler class,
defined for a patch in the Brillouin zone that contains the nodes of pairs of
bands isolated from all other bands by an energy gap Bouhon  _et al._ (2020a).
The patch Euler class is thus computed independently of the potential presence
of nodes connecting the two bands under consideration with other bands outside
the patch.

Dirac Strings
The concrete setup that we will present shortly, however, takes the above
insight to a next level and features a profound role of so-called Dirac
strings in one band gap that affect the charges in the other. Dirac strings
naturally appear when considering smooth representations of real eigenvectors,
marking the necessary discontinuity of the gauge sign upon the creation of any
pair of nodes Ahn  _et al._ (2019). They are the 2D analogue within the real
context of the Dirac strings of 3D Weyl semimetals within the complex context.
It can be shown that by specifying a relative sign as the charge of each node
within a gap (which we represent below through open/filled symbols for
opposite charges and with distinct symbols for different gaps, e.g.,
${\color[rgb]{1,0,0}\boldsymbol{\bullet/\circ}}$, and
${\color[rgb]{0,0,1}\boldsymbol{\blacktriangle/}\triangle}$, for the first and
second gap respectively) as well as the Dirac strings connecting such nodes,
we obtain a picture that exhaustively captures the topological content of the
band structure. To this end only a few rules have to be taken into account in
order to predict the outcomes of the braiding of nodes. Namely, ($i$) each DS
must connect one pair of same-gap-nodes, i.e.
${\color[rgb]{1,0,0}\boldsymbol{\bullet\sim\circ}}$,
${\color[rgb]{0,0,1}\boldsymbol{\blacktriangle\boldsymbol{\sim}}\triangle}$,
${\color[rgb]{1,0,0}\boldsymbol{\bullet\sim\bullet}}$, etc., ($ii$) the charge
of a node switches when crossing the DS of a neighboring (adjacent) gap, i.e.
$[{\color[rgb]{1,0,0}\boldsymbol{\bullet}}{\color[rgb]{0,0,1}\rotatebox[origin={c}]{90.0}{$\boldsymbol{\sim}$}}\times]\leftrightarrow[\times{\color[rgb]{0,0,1}\rotatebox[origin={c}]{90.0}{$\boldsymbol{\sim}$}}{\color[rgb]{1,0,0}\boldsymbol{\circ}}]$,
$[{\color[rgb]{0,0,1}\boldsymbol{\blacktriangle}}{\color[rgb]{1,0,0}\rotatebox[origin={c}]{90.0}{$\boldsymbol{\sim}$}}\times]\leftrightarrow[\times{\color[rgb]{1,0,0}\rotatebox[origin={c}]{90.0}{$\boldsymbol{\sim}$}}{\color[rgb]{0,0,1}\boldsymbol{\triangle}}]$,
($iii$) any couple of Dirac strings in the same gap can be recombined through
the permutation of the end nodes, i.e.
$[{\color[rgb]{1,0,0}\boldsymbol{\bullet_{1}\boldsymbol{\sim}\circ_{2}}}\,{\scriptstyle\cup}\,{\color[rgb]{1,0,0}\boldsymbol{\bullet_{3}\boldsymbol{\sim}\circ_{4}}}]\leftrightarrow[{\color[rgb]{1,0,0}\boldsymbol{\bullet_{1}\boldsymbol{\sim}\bullet_{3}}}\,{\scriptstyle\cup}\,{\color[rgb]{1,0,0}\boldsymbol{\circ_{2}\boldsymbol{\sim}\circ_{4}}}]\leftrightarrow[{\color[rgb]{1,0,0}\boldsymbol{\bullet_{1}\boldsymbol{\sim}\circ_{4}}}\,{\scriptstyle\cup}\,{\color[rgb]{1,0,0}\boldsymbol{\circ_{2}\boldsymbol{\sim}\bullet_{3}}}]$.
We note that these rules are merely different representations of a consistent
charge assignment, and refer to Supplementary Note A for a detailed account on
deforming DS configurations.

Interplay with kagome geometry
The above view relating Euler forms and multi-gap topologies can be made
rigorous using homotopy theory Bouhon  _et al._ (2020b), culminating in more
intricate insulating Euler class topologies. However, it is all we need to put
the physics presented here into perspective. Starting from a kagome geometry,
we combine a symmetry-eigenvalue analysis together with the above new
topological concepts to unravel topological transitions between a rich variety
of phases arising via a symmetry-constrained braiding of band nodes. These
insights are then accompanied by direct signatures that reveal Euler class
physics Ünal  _et al._ (2020).

We now turn to the specific model that will be the center stage to put the
above ideas into play. We depart from $s$-orbitals located at
$(\boldsymbol{r}_{A},\boldsymbol{r}_{B},\boldsymbol{r}_{C})=(\boldsymbol{a}_{1}/2,\boldsymbol{a}_{2}/2,-\boldsymbol{a}_{1}/2-\boldsymbol{a}_{2}/2)$
(see Fig. 2A) and include the NN and the NNN couplings to arrive at the
following Hamiltonian,

$H(\boldsymbol{k})=\left(\begin{array}[]{ccc}\epsilon_{A}&h_{a}(\boldsymbol{k})&h_{b}(\boldsymbol{k})\\\
h_{a}(\boldsymbol{k})&\epsilon_{B}&h_{c}(\boldsymbol{k})\\\
h_{b}(\boldsymbol{k})&h_{c}(\boldsymbol{k})&\epsilon_{C}\end{array}\right),$
(1)

written in the Bloch basis
$\left\\{|\varphi_{i},\boldsymbol{k}\rangle\right\\}_{i=A,B,C}$ using the Zak
gauge (see Supplementary Note B), with

$\begin{array}[]{rcl}h_{a}(\boldsymbol{k})&=&t\cos(k_{1}/2+k_{2}/2)+t^{\prime}\cos(k_{1}/2-k_{2}/2),\\\
h_{b}(\boldsymbol{k})&=&t\cos(k_{2}/2)+t^{\prime}\cos(k_{1}+k_{2}/2),\\\
h_{c}(\boldsymbol{k})&=&t\cos(k_{1}/2)+t^{\prime}\cos(k_{1}/2+k_{2}).\end{array}$
(2)

The wavevector is given over the Brillouin zone in units of the reciprocal
lattice vectors, i.e.,
$\boldsymbol{k}=\frac{1}{2\pi}(k_{1}\boldsymbol{b}_{1}+k_{2}\boldsymbol{b}_{2})$
with $k_{1},k_{2}\in[-\pi,\pi]$ where
$\\{\boldsymbol{b}_{1},\boldsymbol{b}_{2}\\}$ are the reciprocal primitive
vectors dual to $\\{\boldsymbol{a}_{1},\boldsymbol{a}_{2}\\}$, and $t$
($t^{\prime}$) are the NN (NNN) couplings. In the following, we refer to the
$\Gamma$ point as $(k_{1},k_{2})=(0,0)$, the $K=(4\pi/3,-2\pi/3)$ and
$K^{\prime}=(2\pi/3,2\pi/3)$ points, and the $M=(\pi,-\pi)$,
$M^{\prime}=(\pi,0)$ and $M^{\prime\prime}=(0,\pi)$ points.

Figure 2: Multi-gap topology in kagome models. A, Illustration of the kagome
tight-binding model with the NN ($t$) and NNN ($t^{\prime}$) couplings in
arbitrary energy units. B, Band structure of the kagome model with vanishing
onsite energy and $(t,t^{\prime})=(-0.1,-0.3)$. C, Taking
$(\epsilon_{A},\epsilon_{B},\epsilon_{C})=(1,0,-1)$ and $(t,t^{\prime})=(0,0)$
already gives crossing Dirac strings (DS) in both gaps (blue for first gap,
red for second gap). D and E, As the next step, turning off the onsite
potentials and switching on the hopping terms induces band nodes. The nodes in
the second gap (filled/empty red circles indicating $\pm$ topological charges)
cross the DS in the first gap, forming a stable pair as the double node at at
$\Gamma$ (brown circle) in (E) which has finite patch Euler class $\xi=1$.
Meanwhile, the first gap features nodes at $K$ points (triangles). E-K,
Braiding process of the band nodes from one gap to another. Via the outlined
steps depicted in L, band nodes and DS strings evolve such that the degeneracy
at $K$ in the first gap (blue triangles) is tuned into a double node
configuration that has finite patch Euler class in the second gap (brown
circles).

Topological phase diagram
We start from the simplest gapped phase and then introduce band nodes, to
study the evolution and braiding of the nodes along a designed trajectory in
parameter space. To this end, consider the gapped atomic insulating phase that
breaks $C_{6}$ symmetry, by setting
$(\epsilon_{A},\epsilon_{B},\epsilon_{C})=(1,0,-1)$ and
$(t,t^{\prime})=(0,0)$. This system then has the band energies $E_{1}=-1$,
$E_{2}=0$, and $E_{3}=1$, and the corresponding Bloch eigenstates
$|\psi_{1},\boldsymbol{k}\rangle=|\varphi_{C},\boldsymbol{k}\rangle$,
$|\psi_{2},\boldsymbol{k}\rangle=|\varphi_{B},\boldsymbol{k}\rangle$, and
$|\psi_{3},\boldsymbol{k}\rangle=|\varphi_{A},\boldsymbol{k}\rangle$. In the
following, we refer to gap I as the (partial) energy gap between bands 1 and
2, and gap II as the (partial) energy gap between bands 2 and 3.

The Berry phase factors for each band $\mathcal{B}_{n}$ ($n=1,2,3$) over a
non-contractible path
$l_{\boldsymbol{K}}=[\Gamma+\boldsymbol{K}\leftarrow\Gamma]$ where
$\boldsymbol{K}$ is a reciprocal lattice vector, i.e.,
$\exp(\mathrm{i}\gamma_{\mathcal{B}_{n}}[l_{\boldsymbol{K}}])=\langle\psi_{n},\Gamma+\boldsymbol{K}|\prod\limits_{\boldsymbol{k}\in
l}P_{n}(\boldsymbol{k})|\psi_{n},\Gamma\rangle$ with the $n$-th band projector
$P_{n}(\boldsymbol{k})=|\psi_{n},\boldsymbol{k}\rangle\langle\psi_{n},\boldsymbol{k}|$,
are then readily given by

$\begin{array}[]{rcl}\mathrm{e}^{\mathrm{i}\gamma_{\mathcal{B}_{1}}[l_{\boldsymbol{K}}]}&=&\langle\varphi_{C},\Gamma+\boldsymbol{K}|\varphi_{C},\Gamma\rangle=\mathrm{e}^{\mathrm{i}\boldsymbol{K}\cdot\boldsymbol{r}_{C}},\\\
\mathrm{e}^{\mathrm{i}\gamma_{\mathcal{B}_{2}}[l_{\boldsymbol{K}}]}&=&\langle\varphi_{B},\Gamma+\boldsymbol{K}|\varphi_{B},\Gamma\rangle=\mathrm{e}^{\mathrm{i}\boldsymbol{K}\cdot\boldsymbol{r}_{B}},\\\
\mathrm{e}^{\mathrm{i}\gamma_{\mathcal{B}_{3}}[l_{\boldsymbol{K}}]}&=&\langle\varphi_{A},\Gamma+\boldsymbol{K}|\varphi_{A},\Gamma\rangle=\mathrm{e}^{\mathrm{i}\boldsymbol{K}\cdot\boldsymbol{r}_{A}},\end{array}$
(3)

where we have used the periodic gauge for the Wilson loop integration Bouhon
and Black-Schaffer (2017). Berry phases on non-contractible path are also
referred as Zak phases Zak (1989). Taking
$\boldsymbol{K}=\boldsymbol{b}_{1},\boldsymbol{b}_{2}$ and writing the vector
$\boldsymbol{\gamma}_{\mathcal{B}_{n}}=(\gamma_{\mathcal{B}_{n}}[l_{\boldsymbol{b}_{1}}],\gamma_{\mathcal{B}_{n}}[l_{\boldsymbol{b}_{2}}])$,
we find (modulo $2\pi$)

$\begin{array}[]{rclrclrcl}\boldsymbol{\gamma}_{\mathcal{B}_{1}}&=&(\pi,\pi),&\boldsymbol{\gamma}_{\mathcal{B}_{2}}&=&(0,\pi),&\boldsymbol{\gamma}_{\mathcal{B}_{3}}&=&(\pi,0).\end{array}$
(4)

While the total band bundle of the three bands is orientable, i.e., the above
Berry phases sum to $0\>\mathrm{mod}\>2\pi$ in each direction, the ‘flag
bundle’ (where we consider each band separately) is non-orientable as
indicated by the non-zero Berry phases Bouhon  _et al._ (2020b). Each
$\pi$-Berry phase indicates the presence of an odd-number of Dirac strings
crossing perpendicularly the path of integration Ahn  _et al._ (2018). The
corresponding configuration of Dirac strings of this non-orientable atomic
phase is shown in Fig. 2C where the Dirac string across gap I (DSI) is shown
as a blue dashed line, and the Dirac string across gap II (DSII) as a red full
line. It is remarkable that, as long as $C_{2}T$ is preserved, the system
cannot exhibit a fully trivial and orientable band structure, as will become
clear below.

We now deform the model to obtain the simplest kagome model while restoring
the $C_{6}$ symmetry. This is done by setting $t=-1$ and
$\epsilon_{A},\epsilon_{B},\epsilon_{C}\rightarrow 0$. During the deformation
there is a band inversion in each gap, thus creating two simple nodes of
opposite charges in each gap, see Fig. 2D where
$(\epsilon_{A},\epsilon_{B},\epsilon_{C})=(0.5,0,-0.5)$ and
$(t,t^{\prime})=(-0.5,0)$. The final configuration shown in Fig. 2E has two
simple nodal points at the K and $K^{\prime}$ points in gap I (small blue
triangles) that are connected by DSI (a Dirac string in gap I), and a double
nodal point at $\Gamma$ in gap II (large brown circle). DSII is fully
resorbed, and DSI has been chosen as to avoid the nodes of gap II. While these
degeneracies, at $K^{\prime}$s and $\Gamma$, correspond to the two-dimensional
irreducible representations $K_{3}$ and $\Gamma_{5}$ of the elementary band
representation of $s$-atomic orbitals at Wyckoff position $3c$ of layer group
L80 ITC (2006, 2010), our analysis reveals the origin of stability of the
double node at $\Gamma$ which is independent of the crystalline symmetries.
Indeed, in order to go from Fig. 2D to Fig. 2E, DSI must cross one node of gap
II and flip its charge (cf. rule ($ii$) above, and the left column of Fig. S-3
in Supplementary Note A). As a consequence, the two nodes in gap II in the
vicinity of $\Gamma$ have equal charge leading to a patch Euler class of
$\xi=\pm 1$ (equivalently, a frame charge of $-1$), thus indicating that they
cannot be annihilated. A robust associated feature of the nontrivial patch
Euler class of the stable double node degeneracy $\Gamma_{5}$ (at $\Gamma$),
is its quadratic dispersion in all directions, as opposed to the linear
dispersion (“Dirac cone”) around the double degeneracy $K_{3}$ (at $K$), see
Fig. 1C. We emphasize that the pair of nodes at $\Gamma$ are stable upon the
breaking of point group $D_{6h}$ of the kagome lattice (L80) as long as the
$C_{2}T$ symmetry is preserved as predicted by the Euler class (frame charge)
obstruction.

Symmetry-constrained braiding of band nodes
We come to a main feature that concerns the transfer of the double
degeneracies from one gap to the other. We focus here on the $K$ point and
discuss the analog for the $\Gamma$ point in the Supplementary Note C (see
Fig.S-5). These processes thrive on the Kagome symmetries that necessitate
triple degeneracies at $K$, $M$ (see below), and $\Gamma$ in the charge
conversion processes. Namely, we show that the braiding of nodes effectively
acts as a pump of simple nodes (at $K$) and double nodes (at $\Gamma$) from
one gap to the other other, mediated by triply-degenerate points.

The process is illustrated through Figs. 2E-K, while the irreducible
representation (IRREP) content of the band structures is detailed in Fig. S-4
of Supplementary Note C. First, two sextets of nodes in gap II (red circles)
are generated from $\Gamma$, i.e., there are two nodes on each of the three
$M$-$\Gamma$-$M$ and $K$-$\Gamma$-$K^{\prime}$ lines (Fig. 2F). These extra
nodes must appear together upon a band inversion at $\Gamma$ as a result of
the crystalline symmetries ($C_{6v}\subset D_{6h}$) of the system, see Figs.
S-4a-b that show the reordering of IRREPs and the resulting symmetry protected
nodes inside the Brillouin zone. Importantly, the charge configuration
precisely captures the fact that the nodes can be annihilated in the reversed
process. In particular, the Euler class (frame charge) of the nodes at
$\Gamma$ is preserved. Then, the $K$-$\Gamma$-$K^{\prime}$-nodes reach the $K$
points first, thus realizing two triply-degenerate points (represented by the
green stars in Fig. 2G) which is the necessary step upon a band inversion at
$K$ ($K^{\prime}$) (see Figs. S-4a-c). Each triply-degenerate point then gives
birth to one simple node in gap II at the $K$ ($K^{\prime}$) point, and three
simple nodes in gap I (blue triangles) on the $K$-M-$K^{\prime}$ lines (Fig.
2H). This follows by symmetry from the band inversion at $K$ ($K^{\prime}$)
(see Fig. S-4d). We note that the step from Fig. 2G to H requires the
recombination of Dirac strings (cf. rule $(iii)$). Also, defining a loop
around the $K$ ($K^{\prime}$) encircling the $K$-$\Gamma$-$K^{\prime}$-nodes
in Fig. 2F, the frame charge of $\pm k$ is conserved through Fig. 2G and Fig.
2H. Then, the $\Gamma$-$M$-$\Gamma$-nodes in gap II and the
$K$-$M$-$K^{\prime}$-nodes in gap I join at the $M$ points, realizing three
triply-degenerate points (Fig. 2I). Contrary to the previous case, these
triple points are not imposed by symmetry (Figs. S-4d-e) but are rather model
specific (see below). Each triply-degenerate point then gives birth to two
$\Gamma$-$M$-$\Gamma$-nodes in gap I and two $K$-$M$-$K^{\prime}$-nodes in gap
II (Fig. 2J). This results by symmetry from the band inversion at $M$ (Figs.
S-4f). Also, the transition of charges through Fig. 2H-I-J requires the flip
of the charges of the nodes in gap II after crossing the DSI (rule ($ii$)).
The full band structure at this step of the process (Fig.2J) is shown in Fig.
2B. The $K$-$M$-$K^{\prime}$-nodes in gap II then reach the $K$ and
$K^{\prime}$ points, where two double nodes are formed, each with a nontrivial
patch Euler class (frame charge $-1$) (Fig. 2K). We note that the step Fig.
2J-K requires the recombination of DSI (rule ($iii$)).

Key bulk features
Given the complexity of the conversion processes, we recapitulate some of the
essential features lying at the core of the symmetry-constrained braiding for
which symmetry and topology must be married in a new way. In our framework, we
already mentioned the quadratic dispersion in all directions of the double
nodes, e.g. at $\Gamma$ (Fig. S-4a) and at $K^{(\prime)}$ (Fig. S-4g). This is
a robust feature of the non-trivial patch Euler class (frame charge $-1$) of
two stable nodes (see Fig. 1C) that sit on top of each-other as a result of a
crystalline symmetry (i.e. $C_{6}$ at $\Gamma$, and $C_{3}$ at $K^{(\prime)}$)
Lenggenhager  _et al._ (2020).

It is known that three-dimensional semimetals with $\mathcal{PT}$ symmetry and
fourfold or sixfold rotational symmetry can host a double degeneracy at
$\Gamma$ that gives rise to a triply degenerate node along the
$k_{z}$-rotation axis Zhu  _et al._ (2016); Kim  _et al._ (2018); Lenggenhager
_et al._ (2020). This is also another important aspect of the bulk topology
here for the $K^{(\prime)}$ (and $\Gamma$, see SM C) point, that are
unavoidable by symmetry upon the pumping of nodes from one gap to the other.
On the contrary, the triply-degenerate point at the $M$, $M^{\prime}$ and
$M^{\prime\prime}$ points only exist as long as $\boldsymbol{k}$-dependent
same-sub-lattice-site hopping terms are discarded (i.e., $\sim
c_{i,\boldsymbol{k}}^{\dagger}c_{i,\boldsymbol{k}}$, $i=A,B,C$). In other
words, by discarding intra-sublattice-site hoppings the kagome model cannot be
gapped, i.e., the three bands must remain connected through simple, double,
and triple nodes.

Very interestingly, the frame charge associated to one triply-degenerate node
reflects again the dispersion of the bands. A frame charge of $\pm k$ implies
a triple node with two linear bands (at $K$ in Fig. S-4c), while a frame
charge of $-1$ implies a parabolically-dispersed triply-degenerate point (at
$\Gamma$ in Fig. S-5b), as shown in Fig. 1C.

Edge projections
The above determination of bulk charges (i.e., non-Abelian frame charges,
Dirac strings, and patch Euler class) also allows us to systematically obtain
the Zak phases for different edge projections and predict the existence of
edge branches in the different gaps. Indeed, whenever the Zak phase’s
integration path crosses an odd (even) number of Dirac strings, we get a
$\pi$-($0$-)Zak phase. Zak has shown that in one-dimensional quantum systems
(here the 1D slices of the 2D phase) the Zak phase, obtained from the
projection to a group of bands isolated by an energy gap, predicts the
location of the center of charge of the Bloch eigenstates (i.e. the
expectation value of the position operator) Zak (1989). The bulk-edge
correspondence linking the value of the Zak phase with the presence of an edge
state follows from the conservation of charges and the incommensurability
between the number of atomic orbitals, located at the sites that compose the
finite 1D system, and the number of band-projected orbitals originating from
the different gap-separated blocks of bands. Incommensurability, also often
called charge anomaly Su  _et al._ (1979), typically happens when there is a
mismatch between the locations of the atomic and the band orbitals.

We consider the zigzag (see below) and armchair (in Supplementary Note G)
edges, which are defined with respect to the honeycomb structure underlying
the kagome lattice (i.e. connecting the centers of every small triangle
forming the kagome lattice), see the green lines in Fig. 3C and F. The
integration path in momentum space for a given edge geometry is dictated by
the redefinition of the Brillouin zone within the coordinates
$(k_{\parallel},k_{\perp})$ dual to the edge axes $(x_{\parallel},x_{\perp})$,
where $x_{\perp}$ is perpendicular to the edge and $k_{\parallel}$ remains a
good quantum number thanks to the translational invariance along the edge.
Then, $k_{\perp}$ is the integration axis for the Zak phase. The edge-oriented
Brillouin zones are given as the insets of Fig. 5 for the zigzag edges, and of
Fig. S-11 for the armchair edges, where $k_{\perp}$ is the vertical axis in
both cases. Crucially, we find that there is an edge state, given an edge
projection and an energy gap, whenever the Zak phase is zero. This matches
with the fact that the atomic orbitals are located on the boundary of the unit
cell (i.e. Wyckoff position $3c$). In other words, for the given geometry, the
$0$-Zak phase indicates a charge anomaly and hence the existence of a
topological edge state.

From the known bulk charge configuration, we then expect observable
conversions of the edge states through the braiding process. Inversely, the
conversion of the edge states through the different steps of the phase diagram
provides an indirect manifestation of the bulk braiding of nodes. We note in
this regard recent work on one-dimensional gapped real three-band Hamiltonian
realized in electric circuits Guo  _et al._ (2020), where the relation between
Zak phases and non-Abelian charges has been observed. Within this work we
consider genuine Euler class semimetallic materials, a topological phase that
has no one-dimensional equivalent, and the according braiding process (which
similarly requires two spatial dimensions) leading to these phases. Moreover,
the discussed unavoidable bulk features, such as stability of nodes, the
pumping effect and triple degeneracies upon braiding of nodes, are universal
and applicable outside the tunable realm of metamaterials. Indeed, we
emphasize that, while relations between Zak phases and edge states are well-
defined, a general bulk-boundary correspondence in terms of the non-Abelian
frame charges is yet to be formally understood.

Figure 3: Acoustic kagome metamaterials and experimental set-ups. A,
Cylindrical acoustic resonators coupled by a tube connecting them. Upper/lower
panel: Illustration of the coupled resonators (left) and the resultant
hybridized even and odd modes (right) for the NN/NNN coupling.
$l_{1}=18\sqrt{3}$ mm, $l_{2}=54$ mm, $D=19.2$ mm, $H=24$ mm, and
$d_{1}=d_{2}=7$ mm. B, Coupling coefficients $t$ and $t^{\prime}$ vs. the
diameters $d_{1}$ and $d_{2}$. The dashed line and the stars denote the case
in A. C, Photo of a fabricated acoustic metamaterial. Green lines depict the
unit-cell boundaries (lattice constant $36\sqrt{3}$ mm). D and E, The acoustic
band structures of the mematerials with (D) $d_{1}=6.0$ mm and $d_{2}=8.4$ mm
and (E) $d_{1}=0.0$ mm and $d_{2}=9.6$ mm, respectively. F, Photo of the
experimental set-up for the measurements of the bulk and edge band structures.
The upper and lower edges are armchair boundaries, while the left and right
edges are zigzag boundaries. The acoustic source and the detector (a
microphone) are connected to an Agilent network analyzer. The yellow stars
label the positions of the source for the measurement of the zigzag and
armchair edge dispersions.

Acoustic metamaterials
To verify the above evolution and braiding of the band nodes, we created a
physical realization of the kagome model using acoustic metametarials based on
coupled acoustic resonators. Such acoustic metamaterials have been
demonstrated as a highly-tunable and versatile platform for the study of
topological phenomena Yang  _et al._ (2015); Xiao  _et al._ (2015); He  _et
al._ (2016); Lu  _et al._ (2017); Li  _et al._ (2017); He  _et al._ (2018);
Xue  _et al._ (2019); Ni  _et al._ (2019); Zhang  _et al._ (2019). In our
design, each acoustic resonator is a hollow cylinder with an air region of
height $H$ and diameter $D$ encapsulated by a shell made of resin which
confines the acoustic waves. These acoustic resonators play the roles of the
sites in the kagome model. The nearest-neighbor (NN) and next-nearest-neighbor
(NNN) couplings between the acoustic resonators are then achieved through the
air tubes connecting the resonators with diameters $d_{1}$ and $d_{2}$,
respectively. The acoustic metamaterial, with a lattice constant of
$36\sqrt{3}$ mm, is fabricated using the 3D printing technology based on
photosensitive resin (see more details in Supplementary Note D. As illustrated
in Fig. 3A, two acoustic resonators connected with a tube support two
hybridized modes (even and odd modes) with a splitting $\Delta
f=f_{even}-f_{odd}$. The coupling between the two resonators is equal to
$\Delta f/2$. We find that for both the NN and NNN couplings, their strength
can be tuned at will through the diameters $d_{1}$ and $d_{2}$ (Fig. 3B). A
photo of the unit-cell of the fabricated acoustic metamaterial is shown in
Fig. 3C for $d_{1}=6$ mm and $d_{2}=8.4$ mm. The corresponding acoustic band
structure is presented in Fig. 3D, showing the anticipated triple degeneracy
at the M point (similar to the case in Fig. 2I). By tuning the diameters to
$d_{1}=0$ and $d_{2}=9.6$ mm, the acoustic band structure with quadratic band
touching at the $\Gamma$ and M points is realized (see Fig. 3E; similar to the
case in Fig. 2K). Although the acoustic setup does not exactly match the
tight-binding model, the acoustic metamaterial effectively reproduces all
topological band features of the model, as shown below.

Figure 4: Acoustic multi-gap topology of the Euler class. A-F, The acoustic
bulk band structures measured for six kagome metamaterials along the evolution
path illustrated in Figs. 2E, 2G-2K. The white lines represent the calculated
bulk band structures for the six acoustic metamaterials using numerical
methods. Insets show the distributions of the band nodes in gap I and II over
the Brillouin zone for each case. Geometry parameters $d_{1}$ and $d_{2}$ are
shown at the top of each figure. The convention of labeling the band nodes is
the same as in Fig. 2.

Characterization of the bulk bands
The experimental set-up for the measurement of the acoustic band structures is
shown in Fig. 3F. The fabricated acoustic metamaterial has 200 unit-cells,
making a rectangular geometry of $1050$ mm$\times 710$mm. The upper and left
edge boundaries are closed (except at the corners) while the lower and right
edge boundaries are open. Such a structure is optimized for the measurements
of the bulk and edge bands (see Supplementary Note E). For the measurement of
the bulk bands, an acoustic source is placed into a resonator at the center of
the sample to excite the bulk Bloch waves at various frequencies from 5Hz to
3kHz. A probe microphone scans the acoustic pressure at all other resonators.
By Fourier-transforming the detected acoustic pressure distributions at each
excitation frequency, we obtain the acoustic bulk band structure for each of
the six fabricated acoustic metamaterials (see Supplementary Note F for
details). Similarly, when measuring the edge bands, the acoustic source is
placed at the center of the armchair or zigzag edge boundary (labeled by the
yellow stars), while the acoustic pressure distributions along the edge
boundaries are scanned at various excitation frequencies.

We first measure the bulk bands for the six acoustic metamaterials, similar
(but not exactly identical) to the cases in Figs. 2E, 2G-2K along the braiding
trajectory. As presented in Fig. 4, the measured acoustic band structures
agree excellently with the calculated acoustic band structures. These band
structures faithfully regenerate the band nodes evolution in Fig. 2.
Specifically, in Fig. 4A ($d_{1}=8.4$ mm and $d_{2}=0$), the acoustic
metamaterial exhibits the usual dispersion of the kagome lattice with two
Dirac nodes at the K and K′ points in gap I as well as a quadratic band
touching at the $\Gamma$ point in gap II. Another known feature of the
familiar kagome lattice is that the third band emerges as a flat band. In Fig.
4B, by tuning the diameters of the connecting tubes to $d_{1}=7.4$ mm and
$d_{2}=7$ mm, the triple degeneracy emerges at the K and K′ points. Meanwhile,
the Dirac nodes are transferred to gap II in the K-$\Gamma$-K′ lines. In Fig.
4C ($d_{1}=6.8$ mm and $d_{2}=7.6$ mm), the triple degenerate points split
into three Dirac nodes: one at the K (or K′) point in gap II and two on the
K-M-K′ lines in gap I. In Fig. 4D, by tuning the parameters to $d_{1}=6$ mm
and $d_{2}=8.4$, the Dirac nodes in the K-$\Gamma$-K′ lines in gap II and
those in the K-M-K′ lines in gap I coalesce to form the triple degeneracy at
the three M points. Meanwhile, the Dirac nodes in gap II remains at the K and
K′ points. In Fig. 4E ($d_{1}=5$ mm and $d_{2}=9.6$ mm), the triple degeneracy
splits into Dirac nodes again. But now, the Dirac nodes lying on the
K-$\Gamma$-K′ lines are transferred into gap I, while the Dirac nodes on the
K-M-K′ lines are transferred into gap II. The latter Dirac nodes eventually
coalesce at the K and K′ points to form quadratic band touching, as shown in
Fig. 4F by setting $d_{1}=0$ mm and $d_{2}=9.6$ mm. Several remarks are in
place: First, the whole process indicates that the degeneracy at the K point
in gap I switches with the degeneracy on the $\Gamma$-$M$ and $\Gamma$-$K$
lines in gap II. By changing the charges of the nodes in gap II, the
reciprocal braiding processes lead to a double node at the $K$ ($K^{\prime}$)
point in gap II with nontrivial patch Euler class. However, since such
braiding appears six times due to the $C_{6}$ symmetry, the total Euler class
remains nontrivial, $\xi=1$. Second, from material science aspects, except the
band structure in Fig. 3A, all other band structures, particularly the triple
degeneracy in Figs. 3B and 3D and the multiple quadratic degeneracy in Fig.
3F, are unprecedented dispersion features. These exotic band structures reveal
that kagome materials embrace rich physical properties that are attractive for
fundamental science (e.g., kagome Mott insulators) and applications (e.g.,
Dirac materials with tunable bands).

Figure 5: Bulk-edge correspondence in the kagome metamaterials. A-F, Acoustic
dispersions obtained from zigzag edge measurements. The lines represent the
calculated band diagrams for metamaterial structures that finite in the $x$
direction but periodic in the $y$ direction. The inset of each figure depicts
the location of the band nodes in the gap I as well as the Dirac strings. The
projection of the band nodes on the $k_{y}$ axis is shown by the red squares.
The red segments on the $k_{y}$ axis denote the regions with nontrivial Zak
phase. Geometry parameters $d_{1}$ and $d_{2}$ are shown at the top of each
figure. The convention of labeling the bulk band nodes is the same as in Fig.
2.

Measurement of the edge dispersions
The dispersions of the edge states at the armchair and zigzag boundaries of
the six acoustic metamaterials are measured for each acoustic metamaterial. In
Fig. 5, we present the measured results for the zigzag boundaries, while the
experimental data for the armchair boundaries are presented in the
Supplementary Note G. Since gap II is often much smaller than gap I which
hinders the study of edge states in gap II, we focus on the edge states in gap
I in this work. Fig. 5A shows that the edge states emerge only at one side of
the projection of a Dirac node. The zigzag edge states reflect the nontrivial
Zak phase along the $\Gamma$-$K$ direction for each $k_{y}$. The results in
Fig. 5A is consistent with the fact that each Dirac node changes the Zak phase
from $\pi$ to 0 or vise versa. The edge band structure in Fig. 5B reflects
that a triple degeneracy point also flips the Zak phase between $\pi$ and 0.
The triple degenerate point can be regarded as composed of an odd number of
Dirac nodes in both gaps I and II (as seen from Figs. 2F and 2G). Fig. 5C
again reflects that the Dirac nodes (and the triple degenerate points) are the
monopoles of the Berry gauge fields that quantize the Zak phases and therefore
are the terminations of the Dirac strings. From Fig. 5D to Fig. 5F, pairs of
the band nodes project into the same $k_{y}$ position. Therefore, the Zak
phase does not change with $k_{y}$ and the edge states emerge at all $k_{y}$,
as confirmed by consistent results from experiments and simulation. The above
edge phenomena can be understood equivalently through the projection of the
Dirac strings. The $k_{y}$ regions with an even number of projected Dirac
strings correspond to 0 Zak phase, whereas the regions with an odd number of
projected Dirac strings correspond to $\pi$ Zak phase. In our system, because
the Wyckoff position is at the boundary of the unit-cell, the 0 Zak phase
leads to topological edge states, while the regions with $\pi$ Zak phase has
no edge states.

Conclusion and outlook
We theoretically predict and experimentally observe non-Abelian topological
semimetals, a new type of topological phases with multi-gap topological
invariants that depend on Euler class and crystalline symmetries, in tunable
kagome materials. Through the evolution of band nodes and its realization in
acoustic metamaterials, we unveil the general rules for the conversion of band
nodes in different gaps, leading to an exceptionally rich topological phase
diagram. We find that the band evolution processes involve symmetry-enforced
intermediate phases hosting doubly and triply degenerate points with unique
dispersion features that are directly linked to their non-Abelian charges.
Apart from these universal bulk features, we also reveal how the band
evolution process is reflected at the edges through the Zak phase. The
interplay between multi-gap topology, the conversion of non-Abelian
topological charges of nodes between adjacent band gaps, and crystalline
symmetry sets a new benchmark in the study of topological phases of matter.
Moreover, the verification of these unprecedented fundamental phenomena opens
a new frontier of non-Abelian topological semimetals.

Author contributions
A. B. and R.-J. S. performed the theory analysis underpinning the project. B.
J. and J.-H. J. designed the metamaterials. B. J., Z.-K. L., X. Z., B .H., F.
L. and J.-H. J. performed the experiments. A. B., R.-J. S. and J.-H. J. wrote
the manuscript with input from all authors.

Competing interests
The authors declare no competing interests.

Data and materials availability
Relevant data are available in the main text and the supplementary materials.
Additional data is available upon reasonable request.

Acknowledgements
B. J., Z.-K. L. and J.-H. J. are supported by the National Natural Science
Foundations of China (Grant No. 12074281) and the Jiangsu Distinguished
Professor Funding. X. Z. and B. H. are supported by the National Natural
Science Foundation of China (Grant No. 12074279), the Major Program of Natural
Science Research of Jiangsu Higher Education Institutions (Grant No.
18KJA140003). The work at Soochow University is also supported by the Priority
Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.
R.-J. S. acknowledges funding from the Marie Skłodowska-Curie programme under
EC Grant No. 842901 and the Winton programme as well as Trinity College at the
University of Cambridge. F. L. is supported by the Natural Science Foundation
of Guangdong Province (No. 2020A1515010549).

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##  A More on Dirac strings

We here wish to further elaborate on the Dirac strings (DS) as motivated in
the main text. As a first step we can make matters more concrete by showing
how a DS is formed in the context of a simple model. To this end consider a
two-band model $H=h_{x}\sigma_{x}+h_{z}\sigma_{z}$ with
$h_{x}(\boldsymbol{k})=\sin k_{1}$ and $h_{z}(\boldsymbol{k})=1-1/5\cos
k_{2}-t\cos k_{1}$ for $t\in[0,2]$. The two bands are gapped for $0\leq
t<4/5$, then a band inversion occurs at $(k_{1},k_{2})=(0,0)$ for $t\geq 4/5$
from which two simple nodes are produced on the $k_{1}=0$ line that move
outward for increasing $t$, see Fig. S-1. When the nodes annihilate for
$t=6/5$ at the Brillouin zone boundary, the system exhibits a non-contractible
Dirac string at $k_{1}=0$ that can be shown to mark a switch in orientation.

Figure S-1: Dirac string creation in the two-band model (see text). Upon
increasing the control parameter $t$, a pair of nodes is created at the
$\Gamma$-point that move outward and recombine over the edge of the Brillouinz
zone. This thus creates a Dirac string over the $k_{1}$ line for $t>6/5$.

As a next step, we make the recombination rules more insightful. In the main
text we already motivated the rules concerning charge conversion when nodes of
the adjacent gap pass a Dirac string. In addition, we outlined that upon
permuting the end point of a DS, we obtain a set of similar configurations
that merely form different representations of the same system. For
concreteness we recapitulate these charge conversion and recombination rules
in Fig. S-2. The consistency of the underlying physics is effectively
illustrated in Fig. S-3. Using that passing a double Dirac string always gives
a trivial phase factor, owing to the fact that $(-1)^{2}=1$, we see that the
different recombination configurations in the main text can indeed be
adiabatically deformed into each other.

Figure S-2: Schematic illustration of action of Dirac string (DS). The left
column shown the effect of a DS residing in the second gap on nodes in the
first gap. The charge inversion processes of pulling a first-gap node through
this DS or subsequently retracting the string over the other DS that connects
the pair in the first gap is illustrated in the bottom panels, respectively.
The right column depicts the recombination rules of Dirac strings as outlined
in the main text. Every panel is merely a different representation of the same
physics. Figure S-3: Dirac string recombination. As stated in the main text,
the Dirac string (DS) configuration at the first, fourth and final stage are
the same. Deforming DS strings and using that crossing a double DS always
results in a trivial phase factor ($(-1)^{2}=1$), we see that these
configurations indeed amount to the same physics, underpinning the outlined
recombination rules.

##  B The kagome model

We here elaborate more on the proposed model in the main text. We start from a
kagome geometry, see Fig. 2 in the main text. The primitive Bravais lattice
vectors are

$\displaystyle\boldsymbol{a}_{1}$
$\displaystyle=a(3/2\hat{x}+\sqrt{3}/2\hat{y})$
$\displaystyle\boldsymbol{a}_{2}$
$\displaystyle=a(-3/2\hat{x}+\sqrt{3}/2\hat{y}),$
$\displaystyle\boldsymbol{a}_{3}$ $\displaystyle=\hat{z},$

(with the orthonormal Cartesian frame $\langle\hat{x},\hat{y},\hat{z}\rangle$,
e.g., $\hat{x}=(1,0,0)$, $\hat{y}=(0,1,0)$, $\hat{z}=(0,0,1)$), and the
reciprocal primitive vectors are

$\displaystyle\boldsymbol{b}_{1}$
$\displaystyle=2\pi\dfrac{\boldsymbol{a}_{2}\times\boldsymbol{a}_{3}}{\boldsymbol{a}_{1}\cdot(\boldsymbol{a}_{2}\times\boldsymbol{a}_{3})},$
$\displaystyle\boldsymbol{b}_{2}$
$\displaystyle=2\pi\dfrac{\boldsymbol{a}_{3}\times\boldsymbol{a}_{1}}{\boldsymbol{a}_{1}\cdot(\boldsymbol{a}_{2}\times\boldsymbol{a}_{3})},$
$\displaystyle\boldsymbol{b}_{3}$
$\displaystyle=2\pi\dfrac{\boldsymbol{a}_{1}\times\boldsymbol{a}_{2}}{\boldsymbol{a}_{1}\cdot(\boldsymbol{a}_{2}\times\boldsymbol{a}_{3})}.$

The Kagome lattice can be obtained from the layer group LG80, with point group
$D_{6h}=p6/mmm$, and taking the Wyckoff position (WP) $3c$. We define the
position of sub-lattice sites $\\{A,B,C\\}$

$\displaystyle\boldsymbol{r}_{A}$ $\displaystyle=\boldsymbol{a}_{1}/2,$
$\displaystyle\boldsymbol{r}_{B}$ $\displaystyle=\boldsymbol{a}_{2}/2,$
$\displaystyle\boldsymbol{r}_{C}$
$\displaystyle=-(\boldsymbol{a}_{1}+\boldsymbol{a}_{2})/2,$

In the following we assume that a single s-wave orbital is occupied on each
sub-lattice site. The minimal kagome model is then defined as

$H_{\mathrm{K}}(\boldsymbol{k})=\left(\begin{array}[]{ccc}\varepsilon&tc_{2}(\boldsymbol{k})+t^{\prime}c^{\prime}_{2}(\boldsymbol{k})&tc_{1}(\boldsymbol{k})+t^{\prime}c^{\prime}_{1}(\boldsymbol{k})\\\
tc_{2}(\boldsymbol{k})+t^{\prime}c^{\prime}_{2}(\boldsymbol{k})&\varepsilon&tc_{3}(\boldsymbol{k})+t^{\prime}c^{\prime}_{3}(\boldsymbol{k})\\\
tc_{1}(\boldsymbol{k})+t^{\prime}c^{\prime}_{1}(\boldsymbol{k})&tc_{3}(\boldsymbol{k})+t^{\prime}c^{\prime}_{3}(\boldsymbol{k})&\varepsilon\end{array}\right),$
(1)

where we only have included the nearest-neighbor and next-nearest-neighbor
hopping terms, where

$\displaystyle c^{(^{\prime})}_{1}(\boldsymbol{k})$
$\displaystyle=\cos(\boldsymbol{k}\cdot\boldsymbol{\delta}^{(^{\prime})}_{1}),$
$\displaystyle c^{(^{\prime})}_{2}(\boldsymbol{k})$
$\displaystyle=\cos(\boldsymbol{k}\cdot\boldsymbol{\delta}^{(^{\prime})}_{2}),$
$\displaystyle c^{(^{\prime})}_{3}(\boldsymbol{k})$
$\displaystyle=\cos(\boldsymbol{k}\cdot\boldsymbol{\delta}^{(^{\prime})}_{3}),$

with the nearest-neighbor bond vectors

$\displaystyle\boldsymbol{\delta}_{1}$ $\displaystyle=-\boldsymbol{r}_{B},$
$\displaystyle\boldsymbol{\delta}_{2}$ $\displaystyle=-\boldsymbol{r}_{C},$
$\displaystyle\boldsymbol{\delta}_{3}$ $\displaystyle=-\boldsymbol{r}_{A},$

and the next-nearest-neighbor bond vectors

$\displaystyle\boldsymbol{\delta}^{\prime}_{1}$
$\displaystyle=\boldsymbol{r}_{A}-\boldsymbol{r}_{C},$
$\displaystyle\boldsymbol{\delta}^{\prime}_{2}$
$\displaystyle=\boldsymbol{r}_{B}-\boldsymbol{r}_{A},$
$\displaystyle\boldsymbol{\delta}^{\prime}_{3}$
$\displaystyle=\boldsymbol{r}_{C}-\boldsymbol{r}_{B}.$

Let us write the momentum in units of the reciprocal lattice vectors as
$\boldsymbol{k}=k_{1}\boldsymbol{b}_{1}/(2\pi)+k_{2}\boldsymbol{b}_{2}/(2\pi)$,
with $(k_{1},k_{2})\in[-\pi,\pi]^{2}$ inside the Brillouin zone. We then have

$\displaystyle c_{1}(\boldsymbol{k})$ $\displaystyle=\cos[k_{2}/2],$ (2)
$\displaystyle c_{2}(\boldsymbol{k})$ $\displaystyle=\cos[(k_{1}+k_{2})/2],$
$\displaystyle c_{3}(\boldsymbol{k})$ $\displaystyle=\cos[k_{1}/2],$

and

$\displaystyle c^{\prime}_{1}(\boldsymbol{k})$
$\displaystyle=\cos[k_{1}+k_{2}/2],$ (3) $\displaystyle
c^{\prime}_{2}(\boldsymbol{k})$ $\displaystyle=\cos[(k_{1}-k_{2})/2],$
$\displaystyle c^{\prime}_{3}(\boldsymbol{k})$
$\displaystyle=\cos[k_{1}/2+k_{2}],$

which we can readily substitute in $H_{\mathrm{K}}$ Eq. (1).

##  C Band structures and IRREP ordering through the complete braiding process

In conjunction with the detailed enumeration of the charge conversion
processes in the main text (Fig. 2), we here give the band structure
corresponding to each step of the braiding process together with all the
relevant irreducible representations (IRREPs) over the Brillouin zone. This
allows us to relate the movement of the nodes to band inversions at specific
points and regions of the Brillouin zone. By doing so, we have a simple
prediction of the nodes that must (dis-)appear together by symmetry inside the
Brillouin zone, and similarly for the existence of triply degenerate points at
$K$ and $\Gamma$.

Starting with Fig. S-4a) the set of IRREPs correspond to the induced IRREPs
for the elementary band representation of $s$-orbitals maximally localized at
the three sub-lattice sites of Wyckoff position $3c$ for the layer group
$L80$. Since the basal mirror symmetry ($\sigma_{h}:z\rightarrow-z$) does not
play any role here, it is convenient to use the IRREPs of point group $C_{6v}$
(instead of $D_{6h}$) ITC (2006, 2010).

Fig. S-4 provides the data with the IRREP reordering upon the band inversions
associated to the symmetry-constrained braiding of nodes, as discussed in
detail in the main text.

a) $(t,t^{\prime})=(-1,0)$ [E] |  b) $(t,t^{\prime})=(-0.72,-0.28)~{}$[F] |  c) $(t,t^{\prime})=(-0.4,-0.2)$ [G]
---|---|---
|  |
|  |
|  d) $(t,t^{\prime})=(-0.32,-0.23)$ [H] |
|  |
|  |
e) $(t,t^{\prime})=(-0.25,-0.25)$ [I] |  f) $(t,t^{\prime})=(-0.1,-0.3)$ [J] |  g) $(t,t^{\prime})=(t,t^{\prime})=(0,-1)$ [K]
|  |
|  |
Figure S-4: Symmetry enforced nodes inside the Brillouin zone at band
inversions (marked by orange arrows) through the braiding process in the
kagome model. The IRREPs refer to the little co-groups of the high-symmetry
momenta written below. We show the bulk configuration of charges below each
phase for convenience and we give in brackets the label of the corresponding
panel in Fig. 2.

a) $(t,t^{\prime})=(0.2,-0.27)$ |  b) $(t,t^{\prime})=(0.25,-0.25)$ |  c) $(t,t^{\prime})=(0.3,-0.23)$
---|---|---
|  |
|  |
Figure S-5: Continuation of the braiding process of Fig. S-4, here going
through the transfer of the double node at $\Gamma$ (with non-trivial Euler
class) from gap I to gap II, with the appearance of a triply degenerate point.

###  1 Pumping of a double node from gap II to gap I and triply degenerate
point at $\Gamma$

In addition to the braiding process of Fig. 2 (Fig. S-4), we consider the
continuation of the braiding process with the transfer of the double node at
$\Gamma$ from gap II to gap I. As shown in Fig. S-5 following the process of
the main text we may indeed further tune the parameters. Taking opposite signs
between the NN and NNN hopping parameters, and upon increasing $t$ and
decreasing $t^{\prime}$, we see that a characteristic triple band node is
formed at $\Gamma$. First, the transfer of simple nodes in gap II from
$K^{(\prime)}$ to $\Gamma$, and of the simple nodes in gap I to $\Gamma$ as
well (Fig. S-5a) leads to the triple point at $\Gamma$ (Fig. S-5b). Evolving
the parameters further we see the triple node gaps and the double node
degeneracy, quantified by a finite patch Euler class, now resides in the first
gap (Fig. S-5c). This process is readily obtained by symmetry through a band
inversion at $\Gamma$ between the IRRPEs $\Gamma_{1}$ and $\Gamma_{5}$ (Fig.
S-5a-b-c). Remarkably, we can characterize the triple point at $\Gamma$ (Fig.
S-5b), that must necessarily occur upon the band inversion, in terms of its
frame charge.

Given the nontrivial patch Euler class of the double node at $\Gamma$ before
and after the band inversion, the corresponding frame charge is $-1$.
Remarkably, the dispersion of the triple point directly reflects the non-
trivial frame charge $-1$, i.e. there must be two bands from the three with a
quadratic dispersion (see the band structure of Fig. S-5b). This must be
contrasted with the dispersion of the triple point at $K^{(\prime)}$ in Fig.
S-4c) with a frame charge of $\pm k$ where two of the bands must be linear
instead.

We thus conclude the intrinsic relationship between the dispersion at a triply
degenerate point and the value of its frame charge, in the same way as the
patch Euler class dictates the dispersion of a pair of nodes between two
bands.

##  D Construction principles of the kagome acoustic metamaterials

The kagome acoustic metamaterials are constructed using coupled acoustic
resonators. We start from the basic element of the construction, two
identical, cylindrical resonators connected with an air tube in between them
(see Fig. 3A in the main text). The associated effective Hamiltonian (in unit
of frequency) is

$H=\left(\begin{array}[]{cc}\omega_{0}&\kappa\\\ \kappa&\omega_{0}\\\
\end{array}\right)$ (4)

Here, $\omega_{0}$ is the frequency of the lowest resonant modes (s-like
acoustic modes) in the two identical resonators, and $\kappa$ represents a
real-valued coupling coefficient between them. The eigen-frequencies of the
above Hamiltonian are $\omega_{0}\pm\kappa$ associated with the two hybridized
modes. These are the even and odd modes described by the eigenvectors
$1/\sqrt{2}(1,\pm 1)^{T}$, respectively. We use the signs $+$ and $-$ to
represent the even and odd modes, respectively, in analog to the bonding and
anti-bonding states in electronic systems. Therefore, a positive (negative)
coupling coefficient $\kappa$ indicates that the eigen-frequency of the even
mode $f_{\rm even}$ is higher (lower) than that of the odd mode $f_{\rm odd}$.
Quantitatively, $\kappa$ is equal to half of the frequency difference, i.e.,
$\kappa=\frac{1}{2}(f_{\rm even}-f_{\rm odd})$. In our construction, the
geometry parameters of the acoustic resonators are $D=19.2$ mm and $H=24$ mm
for the inner space, while the thickness of the resin is about 2 mm. Such a
design is to ensure that the frequency of the lowest $s$ mode is far away from
the other resonant modes, such as the $p_{z}$ mode (7145.8Hz) and the doubly
degenerate $p_{x}$-$p_{y}$ modes (10470Hz). Note that the lowest $s$ mode is
very special. After hybridization, the even eigen-mode in the coupled two
resonators system always has zero frequency and uniform acoustic pressure
everywhere, as indicated by Fig. 3A in the main text. Nevertheless, such a
zero-frequency even-mode can develop into plane wave like Bloch states in the
lowest acoustic band. We find that as long as the $s$ mode is utilized, the
coupling coefficient $\kappa$ between the resonators is always negative (as
shown in Fig. 3B in the main text).

The kagome lattices are constructed by many coupled acoustic resonators. The
kagome model studied in this work consists of coexisting nearest-neighbor (NN)
and next-nearest-neighbor (NNN) couplings. We realize the NN couplings with
horizontal tubes centered at $z=H/2$ and the NNN couplings with horizontal
tubes either centered at $z=H/4$ or at $z=3H/4$. In this way, all the tubes do
not cross each other and thus remains independent. It is worth noting that the
coupling strengths depend only on the diameters and the lengths of the tubes,
and independent of the z positions of the tubes. Once the lattice constant is
fixed, the lengths of the tubes are determined. The NN and NNN couplings are
then tuned by the diameters of the tubes. The quantitative dependences of the
NN and NNN couplings on the diameters of the connecting tubes are shown in
Fig. 3B in the man text which are obtained from numerical simulations. Here,
we remark that although the acoustic bands cannot be quantitatively explained
by a tight-binding picture Lidorikis  _et al._ (1998), the ability to tune the
NN and NNN couplings at will through the diameters of the connecting tubes
still enable us to reproduce all features of the non-Abelian reciprocal
braiding and the associated band evolution for the lowest three acoustic
bands.

Figure S-6: The designed unit-cell structure. A, The unit-cell of one of the
designed kagome acoustic metamaterial. Cylindrical acoustic resonators are
located at the centers of the unit-cell boundaries (because of the periodicity
of the structure, half cylinders are shown in the figure). The gray regions
represent the photosensitive resin which is used in the 3D printing technology
to fabricate the acoustic metamaterials. Horizontal tubes are used to connect
the acoustic resonators in order to realize the NN and NNN couplings. B, The
geometry parameters of the acoustic kagome metamaterials. The diameters of the
horizontal tubes, $d_{1}$ and $d_{2}$, are given in Figs. 4 and 5 in the main
text, and in Supplementary Fig. S-9 for the six samples. C, Ribbon-like
structure with zigzag boundaries at the left and right sides. D, Ribbon-like
structure with armchair boundaries at the left and right sides. In C and D,
the gray color represents the resin structures. These structures are finite
along the horizontal direction, whereas along the vertical direction, periodic
boundary conditions are used (denoted by the golden lines above and below
these structures). The lengths of these ribbon-like structures are described
by the parameters M and N. In Fig. 5 in the main text and in the Supplementary
Fig. S-11, we take $M=20$ and $N=20$ in the simulation.

##  E Methods and samples

All simulations of acoustic wave dynamics and calculations of acoustic energy
bands were performed using the commercial finite-element solver COMSOL
Multiphysics. The photosensitive resin used in 3D printing can be modelled as
rigid walls for acoustic waves, considering its huge acoustic impedance
contrast to the air background. The systems are filled with air in which the
acoustic waves propagate (modeled with a mass density of 1.300 kg/m3 and the
speed of sound of 343 m/s at room temperatures). In the metamaterials, the
bulk band structures of the acoustic waves are calculated by solving the
eigen-mode equation for acoustic waves in a single unit cell with Bloch
boundary conditions in two-dimensional x-y plane. In comparison, the edge band
structures are calculated by solving the same equation in ribbon-like
structures with the zigzag or armchair boundary in one direction while
periodic boundary condition in the other direction (see next section).
Particularly, in our set-up, the zigzag boundary is along the y direction,
while the armchair boundary is along the $x$ direction (see Fig. 3F in the
main text).

The six experimental samples are all fabricated by commercial 3D printing
technology based on photosensitive resin provided by a company in Shenzhen
city. The fabrication precision is literally 0.1 mm. All samples are
constructed using the coupled acoustic resonators. Each sample has 200 unit
cells and has a geometry of $1050\times 710\times 30$mm3 along the $x$, $y$
and $z$ directions, separately. Each acoustic resonator and each connecting
tube are encapsulated by photosensitive resin with a thickness of 2.1 mm. The
top and bottom resin layers are enhanced to a thickness of 3 mm. A hole is
open at the top of each resonator with a diameter of 7 mm for the detection of
the acoustic wave amplitude in each resonator, $p_{j,\alpha}$ with $j$ and
$\alpha$ labeling the unit cell and the type of sites (A, B or C),
respectively. When detecting the acoustic wave amplitude in the j-th unit cell
at the site, we keep the top holes in all other resonators (except the
resonators with the source and the detector) as closed by rubber plugs.

Figure S-7: Schematic illustration of the experiment setup. The measurement
system includes a speaker as the acoustic source inserted into a resonator to
excite the acoustic waves, a microphone as the acoustic detector connected to
a power amplifier, and an Agilent network analyzer recording the acoustic
signals and analyzing the detected signals at each frequency when the source
frequency sweeps the from 5Hz to 3kHz.

To measure both the bulk and edge dispersions in the same sample, we create a
zigzag boundary at the left edge of the sample and an armchair boundary at the
upper edge, while other edge boundaries are left open for the bulk modes to
leak out of the sample. To disconnect the zigzag and armchair edge boundaries,
we also open the resonators at the corner between the two edge boundaries.
Here, by “open”, we mean that the side surfaces of the resonators are half
open (but the top and the bottom surfaces remain unchanged), so that the
acoustic waves are able to propagate into the free space.

An acoustic source is placed into a resonator at the center of the sample for
the excitation of the bulk modes. The acoustic wavefunction (technically, the
acoustic pressure field) distributions are manually detected for each
resonator through a portable probe microphone with a diameter about 7 mm (just
fitting the size of the top hole in each resonator). The amplitude and phase
distributions of the acoustic wavefunctions (precisely the acoustic pressure
fields) are detected and then automatically recorded by an Agilent network
analyzer (model: Keysight E5061B). By Fourier transforming the detected
acoustic pressure distributions at each excitation frequency, we obtain the
acoustic bulk band structure for each sample (the results are presented in
Fig. 4 in main text). To measure the edge band structures for the zigzag
(armchair) edge boundary, we place the acoustic source into a resonator in the
middle of the zigzag (armchair) boundary.

##  F Details of the acoustic bulk band structure measurements

To increase the acoustic signal resolution, the probe microphone is connected
to a power amplifier with 10 times amplification. A speaker as an acoustic
source is inserted into a resonant cavity through the top hole for the
excitation of the bulk states. The excited acoustic wave is then detected
manually in each resonator using the probe microphone. Technically, the
detected signal is the acoustic pressure in each resonator with both amplitude
and phase signals. Therefore, for each frequency, we obtain a profile of the
detected acoustic field. Some examples of the detected acoustic field profiles
are shown in Supplementary Figure S-8. Note that the measured acoustic field
profiles are different when the acoustic source (labeled as the green star in
the figures) is placed in the A, B or C sites in the same unit cell, since in
kagome lattices, these sites are inequivalent. In particular, for the sixth
sample which has $d_{1}=0$ mm and $d_{2}=9.6$ mm, the kagome lattice can be
divided into two independent parts. The measured acoustic field profiles are
quite different when the position of the acoustic source is different, as
indicated by Supplementary Figs. S-8D-F.

We then carefully check the experimental results by performing the
measurements for each sample with the source in sites A, B and C in the
central unit cell, i.e., we perform three independent measurements. We then
carry out the Fourier transformation of the detected acoustic pressure
profiles on the A, B and C sub-lattice sites and sum the intensity of the
three Fourier transformation signals together. That is, from the detected
real-space acoustic wavefunctions $p_{j,\alpha}$, we obtain the Fourier
transformed signals, $g_{\alpha}({\bf k})=FFT(p_{j,\alpha})$, using the fast
Fourier transformation (FFT) function in Matlab. The quantity presented in
Supplementary Fig. S-9 is $P({\bf k})=\sum_{\alpha}|g_{\alpha}({\bf k})|^{2}$.
Specifically, we present for each sample the obtained $P({\bf k})$ for three
different positions of the acoustic source: with the source in sites C (left
column), B (middle column) and A (right column) in the central unit cell. The
obtained $P({\bf k})$’s are slightly different for the three source positions.
For some samples, such differences (essentially due to the fact that the A, B
and C sites are inequivalent in kagome lattices) are more noticeable, which
may be due to the properties of the bulk eigen-states of the measured energy
bands. Nevertheless, the overall results show excellent consistency with each
other and with the calculated bulk band structure (denoted by the white
curves). For each sample, we adopt the $P({\bf k})$ which has the best
spectral resolution and/or the best consistency with the calculated acoustic
bulk band structures, since each $P({\bf k})$ is a legitimated experimental
representation of the acoustic bulk band structure for one sample. The adopted
$P({\bf k})$ in Fig. 4 in the main text are labeled by the dashed boxes in
Supplementary Fig. S-9 for each of the six fabricated samples (Each sample
corresponds to one row in the figure). Finally, we remark that because of the
low excitation efficiency of the speaker at low frequencies (limited mainly by
the size effect), the detected acoustic signals at low frequencies are much
weaker and so are the Fourier transformed signals $P({\bf k})$ at low
frequencies, as indicated by Fig. 4 in the main text and Supplementary
Fig.S-9.

Figure S-8:  Examples of the measured acoustic pressure profiles in the bulk.
A-C, Measured acoustic pressure profiles of the fifth kagome metamaterial
(corresponding to Fig. 4E in the main text). D-F, Measured acoustic pressure
profiles of the sixth kagome metamaterial (corresponding to Fig. 4F in the
main text). The excitation frequency, 1705 Hz, is labeled in each figure. The
sample parameters, $d_{1}$ and $d_{2}$, are given at the top of the Figures D
and F. The first row (i.e., A and D): the detected acoustic signal profiles
when the source is placed at the A site. The second row (i.e., B and E): the
detected acoustic signal profiles when the source is placed at the B site. The
third row (i.e., C and F): the detected acoustic signal profiles when the
source is placed at the C site. The color scale represents the absolution
value of the acoustic pressure. Figure S-9:  Experimentally measured acoustic
bulk bands. Each row corresponds to a sample. The left, middle and right
columns correspond to the cases with the acoustic source placed at the C, B
and A sites, respectively. The sample parameters, $d_{1}$ and $d_{2}$, are
given at the top of the figures in the left column. Figures with a dashed box
are selected into Fig. 4 in the main text.

##  G Details of the acoustic edge dispersion measurements

To measure the acoustic edge dispersions, we insert the acoustic source (i.e.,
the speaker) into a resonator in the middle of the zigzag (armchair) boundary
and detect the acoustic pressure profile along the zigzag (armchair) edge.
Examples of the measured acoustic pressure profiles along the edges for each
sample are presented in Supplementary Fig. S-10 for both the armchair edges
(Figs. 5A-F) and the zigzag edges (Figs. 5G-L).

Figure S-10:  Observation of topological edge states in finite-sized acoustic
kagome metamaterials. A-F: The measured acoustic pressure profiles along the
armchair edge when the source (labeled by the green star in each figure) is
placed at the middle of the armchair boundary. G-L: The measured acoustic
pressure profiles along the armchair edge when the source (labeled by the
green star in each figure) is placed at the middle of the zigzag boundary. The
geometry parameters of the sample and the excitation frequency of the source
are given at the top of each figure. Acoustic pressure profiles at a few
layers of resonators near the edge boundaries are measured in the experiments,
which is found to be sufficient to extract the acoustic edge dispersions.

In the main text, we present the measured acoustic dispersions for the zigzag
edge boundaries in Fig. 5. We focus on the edge states in gap I since it is
much larger than gap II, and the edge states in gap I is more accessible in
experiments in most of the six acoustic kagome metamaterials. In addition, we
show in the main text that the topological edge states demonstrate the bulk-
edge correspondence as dictated by the band nodes and the Dirac strings. In
particular, for the wavevector regions with an even number of projected Dirac
strings, topological edge states emerge. In contrast, if there are an odd
number of projected Dirac strings, there is no edge state.

Here, we present the measured acoustic edge dispersions in Supplementary Fig.
S-11 and discuss the underlying mechanisms and the bulk-edge correspondence.
As shown in Supplementary Fig. S-11A, for the armchair boundary, the Dirac
nodes at both the $K$ and $K^{\prime}$ points are projected into the $k_{x}=0$
point. According to the bulk-edge correspondence, topological edge states
emerge for all $k_{x}$. This is essentially because the overlapped projection
of the Dirac nodes does not change the Zak phase. Besides, the Dirac strings
are projected even times onto the $k_{x}$ axis. These observations lead to the
conclusion that there should be edge states for all $k_{x}$. Similar
conclusions can be arrived at for Supplementary Figs. S-11B-D, where edge
states are found for all $k_{x}$. In Supplementary Fig. S-11E, according to
the same bulk-edge correspondence, we find that the edge states emerge for
large positive and negative $k_{x}$’s. However, for the sample in
Supplementary Fig. S-11F, because the NN coupling vanishes, the system breaks
into two independent parts. Each of them can be regarded as a triangular
lattice. Therefore, additional trivial boundary states can be triggered for
small $|k_{x}|$ between the projections of the Dirac nodes.

For all six samples, we find consistency between the calculated and measured
edge dispersions. The slight deviations between the experimental data and the
theoretical curves are probably due to the geometry imperfection of the
resonators near the edge boundaries in some samples due to uncontrollable
fabrication errors.

Figure S-11: Measured acoustic edge dispersions along the armchair boundary.
A-F: Acoustic dispersions measured along the armchair boundary. White curves:
calculated bulk dispersions. Green curves: calculated edge dispersions in gap
I. The right-side panel in each figure depicts the band nodes and the Dirac
strings in gap I. The projection of the band nodes on the axis is shown by the
red squares. The red segments on the axis denote the regions with nontrivial
Zak phase that predicts the emergence of the topological edge states.