# Experimental observation of meronic topological acoustic Euler insulators

Bin Jiang  These authors contributed equally  Institute of Theoretical and
Applied Physics, School of Physical Science and Technology & Collaborative
Innovation Center of Suzhou Nano Science and Technology, Soochow University,
Suzhou 215006, China  Adrien Bouhon  These authors contributed equally
Corresponding author<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United
Kingdom  Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden
Shi-Qiao Wu  These authors contributed equally  Ze-Lin Kong  These authors
contributed equally  Zhi-Kang Lin  Institute of Theoretical and Applied
Physics, School of Physical Science and Technology & Collaborative Innovation
Center of Suzhou Nano Science and Technology, Soochow University, Suzhou
215006, China  Robert-Jan Slager  Corresponding author<EMAIL_ADDRESS>These
authors contributed equally  TCM Group, Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Jian-Hua
Jiang  These authors contributed equally  Corresponding author:
<EMAIL_ADDRESS>Institute of Theoretical and Applied Physics, School
of Physical Science and Technology & Collaborative Innovation Center of Suzhou
Nano Science and Technology, Soochow University, Suzhou 215006, China  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

###### Abstract

We report on the experimental observation of a gapped nontrivial Euler
topology in an acoustic metamaterial. The topological invariant in this case
is the Euler class, which characterizes the two-dimensional geometry of the
acoustic Bloch wave function of multiple bands in the Brillouin zone—a feature
that cannot be captured by conventional topological classification schemes.
Starting from a carefully designed acoustic system, we retrieve a rich phase
diagram that includes the gapped Euler phase on top of non-Abelian topological
nodal phases. In particular we find that the gapped phase is characterized by
a meron (half-skyrmion) type topology in which the cancellation of the orbital
Zak phases by Zak phases of the bands contributes in forming the
characterizing winding number of the acoustic bands. Using pump-probe
techniques, we are able to extract the wave functions of the acoustic Bloch
bands and demonstrate the non-trivial Euler topology via a direct observation
of the meron geometry of these bands in the Brillouin zone, in addition to the
detection of the acoustic bulk and edge dispersions. We finally observe a
topological edge state in the gapped Euler phase which is a direct
manifestation of the underlying cancellation of the Zak phases. These
experimental features reveal the unprecedented topological Euler phase as well
as setting a benchmark for probing acoustic wave function geometry in the
Brillouin zone.

Introduction.— Topological phases of matter Qi and Zhang (2011); Hasan and
Kane (2010) offer an intriguing regime with rich unprecedented phenomena.
Although the past decade has seen remarkable theoretical progresses Fu (2011);
Slager  _et al._ (2013); Bouhon and Sigrist (2014); Soluyanov and Vanderbilt
(2012); Wieder and Bernevig (2018); Cornfeld and Carmeli (2021); Van Mechelen
_et al._ (2022); Turner  _et al._ (2012); Schrunk  _et al._ (2022); Shiozaki
and Sato (2014); Slager  _et al._ (2014); Armitage  _et al._ (2018); Scheurer
and Slager (2020); Ahn and Yang (2019); Alexandradinata and Bernevig (2016);
Chiu  _et al._ (2016), culminating in versatile topological classification
schemes Kruthoff  _et al._ (2017); Bouhon and Black-Schaffer (2017); Po  _et
al._ (2017); Bradlyn  _et al._ (2017); Bouhon and Black-Schaffer (2017). These
methods by and large can be traced back to evaluating symmetry representations
of all the bands below an energy gap. Recently, however, a class of
topological phases beyond such a scheme has been steadily attracting attention
Wu  _et al._ (2019); Bouhon  _et al._ (2020a); Ahn  _et al._ (2019). These
phases instead depend on the wave function geometry of the bands separated by
multiple band gaps Wu  _et al._ (2019); Bouhon  _et al._ (2020b). Under the
protecting symmetry, the nontrivial geometry of the multi-band wave functions
is akin to dislicinations or vortices in bi-axial nematic liquid crystals and
are characterized by the non-Abelian topological charges Nissinen  _et al._
(2016); Alexander  _et al._ (2012); Liu  _et al._ (2016); Volovik and Mineev
(2018); Beekman  _et al._ (2017). Such topology resides between multiple
bands, rather than within the bands below a single energy gap. As a
consequence, the topological invariant is not only determined by the band gap
opening scenario, but also by the braiding of band degeneracies among
different bands. A prototype example of such non-Abelian topology is the Euler
class that acts as the paradigmatic real-valued analog of the Chern number and
signifies that band degeneracies with the same non-Abelian charge cannot be
annihilated in pairs. Generally, such multi-gap phases can be analyzed using
homotopy arguments Bouhon  _et al._ (2020b) and are increasingly related to
emergent physical effects and retrieved in various experimental contexts. For
instance, it was predicted that upon quenching a system with a non-trivial
Euler Hamiltonian, monopole-antimonopole pairs can form in the Brillouin zone
Ünal  _et al._ (2020). This phenomenon was recently observed in trapped ion
experiments Zhao  _et al._ (2022). Similarly, Euler class and non-trivial
braiding was predicted in phononic systems Peng  _et al._ (2022a, b); Park
_et al._ (2021); Lange  _et al._ (2022) and electronic systems that are
strained Bouhon  _et al._ (2020a); Könye  _et al._ (2021), that undergo a
structural phase transition Chen  _et al._ (2022); Bouhon  _et al._ (2021), or
are submitted to an external magnetic field Guan  _et al._ (2021). The most
immediate playground for these uncharted topological phases of matter is
however the context of metamaterials Guo  _et al._ (2021); Park  _et al._
(2022a); Jiang  _et al._ (2021); Qiu  _et al._ (2022); Park  _et al._ (2022b);
Ezawa (2021). The non-Abelian topological charges were recently detected in
one-dimensional (1D) electrical circuit metamaterials Guo  _et al._ (2021).
Meanwhile, in two-dimensional (2D) acoustic metamaterials, the braiding and
non-Abelian topological phase transitions were demonstrated Jiang  _et al._
(2021). These developments strongly indicate the potential and rapid
development of this new multi-gap topological field.

Here, we report on the experimental observation of topological Euler insulator
phase in an acoustic metamaterial realization. By designing a kagome system
with up to the third nearest-neighbor (TNN) hoppings, we are able to create a
topological band gap with nontrivial Euler class that can appear only in two
spatial dimensions. Due to the interplay of the nearest-neighbor (NN), the
next nearest-neighbor (NNN), and the TNN couplings, we find a rich phase
diagram where the gapped Euler phase and other non-Abelian topological nodal
phases emerge. For instance, the gapped Euler phase is featured with the meron
(half-Skyrmion) pattern of the single isolated bulk wave function in the
reciprocal space. The meron number in the Brillouin zone gives exactly the
Euler class topological index ($\chi\in\mathbb{Z}$). We theoretically show an
intricate cancellation of bulk Zak phases in the gapped Euler phase that
permits an odd Euler class ($\chi=1$) which explains the underlying meronic
structure. The cancellation of the Zak phases is further pinned by the direct
observation of an edge state corresponding the updated bulk-edge
correspondence. Using acoustic metamaterial realization and pump-probe
measurements, we manage to observe the topological Euler insulator phase by
retrieving the bulk and edge dispersions as well as verifying the acoustic
meron pattern in the reciprocal space. These experimental discoveries are
enabled by the state-of-the-art pump-probe measurements facilitated by the
spectral decomposition of the response functions which set benchmark in the
experimental detection of acoustic topological phenomena. Our study paves the
way toward topological Euler insulators and related non-Abelian topological
phases with emergent physical properties.

Figure 1: (a) A kagome tight-binding model with the unit-cell boundaries
delineated by the dashed lines. The lattice vectors, a1 and a2, are denoted by
the green arrows. (b) The unit-cell structure. The NN, NNN, and TNN couplings
are denoted as $t$, $t^{\prime}$ and $t^{\prime\prime}$, respectively (colored
lines). The three inequivalent sublattice sites A, B, and C are labeled by the
black dots. (c) The 3D bulk dispersion of a topological Euler semimetal with
$t=t^{\prime}=-1$ and $t^{\prime\prime}=0$. The right panels give the
Brillouin zone [the gray hexagon region in the lower inset where we also give
the coordinate system of the wavevector space used in this work ($k_{1}$ and
$k_{2}$)] and the band degeneracy points (the upper inset) where the color map
indicates the energy difference between the second and the third bands. (d)
The 3D bulk dispersion of a topological Euler insulator with $t=t^{\prime}=-1$
and $t^{\prime\prime}=-0.8$. The triple points are gapped and become Dirac
points in gap II (see right inset).

Model setting for topological Euler phases.— We consider a three-band tight-
binding model in 2D kagome lattice with the NN, NNN, and TNN couplings. There
are three inequivalent lattice sites in each unit-cell, labeled separately as
A, B, and C. These sites are coupled to each other with the NN, NNN, and TNN
couplings, denoted as $t$, $t^{\prime}$ and $t^{\prime\prime}$, respectively
[Figs. 1(a) and 1(b)]. We emphasize that the inclusion of the TNN couplings,
which is absent in Ref. Jiang  _et al._ (2021), is crucial for the emergence
of the topological Euler insulator phase. In the presence of both the
inversion (${\cal P}$), or equivalently the $180^{\circ}$ rotation ($C_{2z}$),
and time-reversal (${\cal T}$) symmetries, once a proper basis is chosen, the
Hamiltonian matrix and the its eigenvectors are real-valued for any wavevector
$\boldsymbol{k}$ in the Brillouin zone Bouhon  _et al._ (2020a). The explicit
Hamiltonian matrix and the Bloch bands (their dispersions and wave functions)
are given in Supplemental Material SM (2022). This model hosts a rich
topological phase diagram that can be directly probed in an acoustic
metamaterial setting Bouhon  _et al._ (2020a). Here, we show two prototype
phases in Figs. 1(c) and 1(d). Figure 1(c) gives a non-Abelian topological
semimetal phase with $t^{\prime\prime}=0$ where the three bands are
interconnected. There are triple points (green dots) as the linear crossing
points between the three bands (at the $M$ and equivalent points) as well as
two Dirac points (at the $K$ and $K^{\prime}$ points, labeled by the red dots)
and a quadratic point (the red triangle) in the partial gap between the second
and third bands (denoted as gap II). Notably, the triple points play a crucial
role in the braiding, transfer and merging of band nodes in gaps I and II
Jiang  _et al._ (2021). The topological semimetal phase with triple points
serve as the unavoidable intermediate states for the evolution of various non-
Abelian topological semimetal phases. Without the TNN couplings, the
topological semimetals cannot be gapped, i.e. the three bands remain
connection via nodal points in both partial gaps. In contrary, with finite TNN
couplings, a gap between the first and second bands can be introduced and a
topological Euler insulator phase emerge as illustrated in Fig. 1(d).

This topological gapped Euler phase is characterized by several salient
features. First, the two-dimensional topology of real band structures is
characterized by the Euler class $\chi\in\mathbb{Z}$, whenever we can separate
a two-band subspace from all the other bands, as is the case here. This
topological invariant is most conveniently obtained as the winding of the two-
band Wilson loop Bouhon  _et al._ (2019); Ahn  _et al._ (2018) computed with
the two Bloch eigenvectors of the subspace SM (2022). Scanning the whole
Brillouin zone, we find that the Wilson loop spectrum is gapless, with each
Wilson branch (i.e. the phases of the Wilson loop eigenvalues) winding by
$2\pi$ in opposite directions [see Fig. 2(b)]. Interestingly, the data of the
crystal symmetries of the gapped kagome phase can be advantageously used to
predict the symmetry protected winding of the Wilson loop, thus obstructing a
trivial topology with no winding. This is most directly shown by considering
the flow of Wilson loop over a symmetric patch that covers one sixth of the
Brillouin zone, obtained by deforming the path $l_{\Gamma K}$ into $l_{\Gamma
M}$ (see Fig. 7(a) in SM (2022)). It was shown in Ref. Bouhon  _et al._
(2019); Höller and Alexandradinata (2018) that the hexagonal symmetries with
time reversal symmetry imposes a (minimal) finite quantized winding of the
Wilson loop over the patch of $|\pi-2\pi/3|=\pi/3$ for each Wilson branch, see
Fig.2 (c), leading to a total winding of $6\pi/3=2\pi$ [Fig. 2(b)]. This
fragile crystalline topology is actually the generic nature of the split
elementary band representation of the kagome lattice Bradlyn  _et al._ (2017);
Bouhon  _et al._ (2019), see SM (2022) for a more detailed discussion.

Figure 2: (a) The vector field formed by the Bloch wave function of the first
band in the Brillouin zone for the topological gapped Euler phase in Fig.
1(d). Green arrows indicate $k_{1}$ and $k_{2}$. (b) The Wilson loop spectrum
of the two bands above the gap (red lines) and the single band below the gap
(blue line). (c) Quantized Wilson loop flow over a symmetric patch of the
Brillouin zone dictated by the hexagonal symmetries: the base path at $t=0$ is
$l_{\Gamma K}=[\Gamma\rightarrow
K\rightarrow\Gamma+\mathbf{b}_{1}-\mathbf{b}_{2}]$, and at $t=1$ it is
$l_{\Gamma M}=[\Gamma\rightarrow
M\rightarrow\Gamma+\mathbf{b}_{1}-\mathbf{b}_{2}]$. The quantization is a
consequence of the splitting of the elementary band representation of the
kagome lattice (see text and SM (2022)). (d) Energy spectrum of the zigzag
structure based on the same tight-binding model. Topological edge states in
the gap are highlighted by the blue line.

While the hexagonal symmetries relate to the winding of Wilson loop, we are
here actually interested in the more fundamental essence of the Euler topology
which does not depend on the hexagonal symmetries and thus goes beyond the
symmetry-indicated topologies in general. Namely, we demonstrate, in theory
and in experiment, that the system exhibits an intrinsic nontrivial meronic
geometry. This winding is robust even when the hexagonal symmetries are broken
(see SM (2022)).

Remarkably, we find that the Bloch eigenvector
$u_{1,\boldsymbol{k}}\in\mathbb{R}^{3}$ (written in the
$\\{A,B,C\\}$-sublattice basis) of the first band fully captures the
nontrivial Euler topology of the two-band subspace. Indeed, the Euler class of
a three-band gapped Euler phase can be readily expressed in term of the
Pontryagin number Bouhon  _et al._ (2020a); Ünal  _et al._ (2020) of
$u_{1,\boldsymbol{k}}$. This is a direct consequence of the fact that the
three eigenstates form a dreibein or orthonormal frame Bouhon  _et al._
(2020a); Ünal  _et al._ (2020).

In our system the vector field of the Bloch eigenvector of the first band is
in fact even more special and emerges as a meron pattern (half of a Skyrmion)
in the Brillouin zone, as shown in Fig. 2(a) [the green arrows give a choice
of reciprocal primitive vectors, see the right column of Fig. 1]. In fact, the
meron number of this vector field is in direct correspondence with the Euler
class $\chi=1$ of the gapped phase (see Supplemental Material SM (2022)). This
specific geometric structure can only be understood by considering that the
orbital $\pi$-Zak phases, originating from the off-centering of the sublattice
sites of kagome, are cancelled by $\pi$-windings of the Bloch eigenwave
function, i.e. note the $\pi$-winding of the vector field along the green
arrows in Fig. 2(a) 111We remind that a Zak phase is a Berry phase computed
over a non-contractible path of the Brilouin zone, i.e. a path along a
periodic cycle of the Brillouin zone.. (We give a very detailed analysis of
the cancelling of the Zak phases in SM (2022).) From a more mathematical point
of view this means that the system, which hosts a non-orientable flag atomic
limit 222This limit is obtained e.g. by setting all hopping parameters to zero
and imposing different onsite energies for each orbital,
$\varepsilon_{A}\neq\varepsilon_{B}\neq\varepsilon_{C}$. The Zak phase factors
along a path $\boldsymbol{k}\rightarrow\boldsymbol{k}+\boldsymbol{K}$, with
the reciprocal lattice
$\boldsymbol{K}=n_{1}\boldsymbol{b}_{1}+n_{1}\boldsymbol{b}_{2}$, are then
readily given by $e^{\text{i}\boldsymbol{K}\cdot\boldsymbol{r}_{\alpha}}$ for
$\alpha=A,B,C$. (where every atomic band exhibits $\pi$-Zak phases indicating
non-trivial first Stiefel-Whitney class), is converted into an orientable
gapped Euler phase where all Zak phases are zero. This is to be contrasted
with the three-band gapped Euler phases considered previously Bouhon  _et al._
(2020b); Bzdušek and Sigrist (2017) that are all connected to an orientable
flag atomic limit instead, leading to the constraint of an even Euler class,
leading to a vector field geometry for these phases that must correspond to an
integer multiple of Skyrmions. We conclude this more conceptual part by
reminding that the Euler class indicates the minimal number $\\#=|2\chi|\in
2\mathbb{N}$ of nodal points formed by the two bands that cannot be
annihilated as long as the band gaps above and below the bands remain open Ahn
_et al._ (2018); Bouhon  _et al._ (2019); Ahn  _et al._ (2019); Bouhon  _et
al._ (2020a). It turns out that the gapped kagome phase hosts a stable pair of
nodal points merged at $\Gamma$ (realizing the two-dimensional irreducible
representation (IRREP) $\Gamma_{5}^{+}$ of the point group $D_{6h}$),
accompanied by simple nodes at K and M.

We finally note that an edge band emerges in the topological gap of the Euler
phase at the zigzag edges as shown in Fig. 2(d). This is a direct consequence
of the cancellation of the Zak phases in the gapped Euler phase. (Indeed, the
bulk-edge correspondence relates the existence of the edge state for a zero
Zak phase when the atomic orbitals are off-centered in the unit cell, as is
the case for the kagome lattice.) We thus conclude that the edge state is an
indirect signature of the topological odd Euler phase of the gapped kagome
system. (Edge states at the armchair edges are discussed in Supplemental
Material SM (2022).)

To verify the above unconventional features in experiments, we design and
fabricate an acoustic metamaterial to yield an acoustic topological Euler gap
[Fig. 3(a)]. The unit-cell structure is illustrated in the insets with a
lattice constant $a=36\sqrt{3}$ mm. There are three cylindrical acoustic
cavities (labeled as A, B, and C) in a unit-cell, representing three sites in
the unit-cell of the tight-binding model in Fig. 1. The NNN and TNN couplings
are realized by the tubes (with radii $r_{1}$ and $r_{2}$, respectively)
connecting these cavities. Note that the tubes for the NNN couplings are
realized by two layers to enforce the inversion symmetry. The tubes realizing
the TNN couplings intersect at the unit-cell center which also provide the NN
couplings through indirect process. By tuning the radii of these tubes, we can
realize the topological gapped Euler phase in the acoustic bands. The bulk
acoustic bands from both simulation and experiments are presented in Fig.
3(b). The simulation is based on the acoustic wave equation using commercial
finite-element simulation methods (see Supplemental Material SM (2022)). The
experiments are based on acoustic pump-probe measurements. Specifically, an
acoustic source is placed at the center of the system (indicated by the blue
star) to excite the bulk Bloch acoustic waves. A detector (i.e., a small
microphone) is used to scan the acoustic wave function in the whole system. By
varying the excitation and detection frequency (they are always fixed to be
equal), we obtain the acoustic wave functions at different frequencies. Upon
Fourier transformation of the probed wave functions, we obtain the dispersions
of the excited bulk Bloch acoustic waves. The measured dispersions of the
acoustic bulk Bloch states agree well with the simulated acoustic dispersions.
These acoustic band structures all show a band gap between the first band the
remaining bands. Besides, there are five Dirac points and a quadratic point in
the partial gap II which have a total patch Euler class of 1, indicating the
nontrivial topological Euler class index of this gapped phase. In Supplemental
Material SM (2022), we also give the measured dispersions around these
degeneracy points.

Next we measure the dispersion of the acoustic edge states. By placing the
acoustic source at the center of an zigzag edge [the red triangle in Fig.
3(a)], we are able to excite the edge states within the bulk band gap and
measure their dispersion using similar pump-probe techniques (i.e., scanning
the acoustic wave functions along the zigzag edge at various excitation
frequencies and then performing the Fourier transformation of the detected
wave function along the edge direction). As shown in Fig. 3(c), the measured
edge dispersion is comparable with the simulated dispersion (the yellow
curve), confirming the emergence of the edge states in the band gap. We
emphasize that although the acoustic band below the gap has vanishing Berry
phase, the edge states emerge due to the following reasons. First, the edge
states is a manifestation of the fragile topology of the second and third
bands [see Figs. 2(b) and 2(c)] Bouhon  _et al._ (2019); Alexandradinata and
Bernevig (2016). Second, as mentioned above, due to the fact that the sites
are located at the unit-cell boundaries, the trivial Zak (Berry) phase yields
topological edge states.

Figure 3: (a) Photograph of the fabricated acoustic metamaterial that realizes
topological Euler band gap. Upper right inset: the unit-cell structure
(geometric details are given in Supplemental Material). Lower right inset:
zoom in photo of one unit-cell. unit-cell boundaries also delineate dashed
lines. (b) Acoustic bulk band structures from simulations (yellow curves) and
experiments (color map). (c) Measured (color map) and simulated (yellow
curves) dispersions of the acoustic zigzag edge states. White curves represent
simulated bulk bands projected to the zigzag edge.

We now reveal the most salient feature of the topological Euler gap, the meron
pattern of the eigenvector in the Brillouin zone. For this purpose, we develop
a technique of spectral decomposition based on the acoustic pump-probe setup.
The underlying scheme is based on the fact that the two-points pump-probe
signal gives the Green’s function of the acoustic waves. At a frequency for
excitation and detection, $f$, the two-points Green’s function is a $3\times
3$ tensor since the source and detector can be at the A, B, and C sublattice
sites in different unit-cells,
$\chi_{\beta\alpha}(\boldsymbol{r}_{s},\boldsymbol{r}_{d},f)$ where
$\beta,\alpha=(A,B,C)$ denote the pumping and detection sites, respectively,
$\boldsymbol{r}_{s}$ ($\boldsymbol{r}_{d}$) denotes the position vector of the
source (detector). Upon Fourier transformation of the Green’s function, the
measured response function is $\chi_{\beta\alpha}(\boldsymbol{k},f)$.
Theoretically,

$\chi_{\beta\alpha}(\boldsymbol{k},f)\sim\sum_{n}\frac{u_{n\boldsymbol{k}}^{\beta}u_{n\boldsymbol{k}}^{\alpha~{}\ast}}{f-(f_{n\boldsymbol{k}}+i\gamma_{n\boldsymbol{k}})},$
(1)

where $n=(1,2,3)$ is the band index, $f_{n\boldsymbol{k}}$ and
$\gamma_{n\boldsymbol{k}}$ are the eigenfrequency and the broadening of the
Bloch bands. $u_{n\boldsymbol{k}}$ is the eigenvector of the Bloch states of
the $n$-th band at the wavevector $\boldsymbol{k}$ written in the local basis
of the sublattice sites A, B, and C. To measure the acoustic Bloch eigenstates
$u_{1\boldsymbol{k}}$ of the first band, we first measure the response
function $\chi_{\beta\alpha}(\boldsymbol{k},f)$ through the pump-probe
technique and the Fourier transformation. We then examine the response
function at resonance with the first acoustic band, $f=f_{1\boldsymbol{k}}$,
where the dominant contribution in the above equation comes from the first
band. We use a singular value decomposition of the tensor
$\chi_{\beta\alpha}(\boldsymbol{k},f_{1\boldsymbol{k}})$ to extract this
dominant contribution. This process also gives the eigenstates vector
$u_{1\boldsymbol{k}}$ as the normalized eigenvector associated with the
largest singular value of the $3\times 3$ response tensor
$\chi_{\beta\alpha}(\boldsymbol{k},f_{1\boldsymbol{k}})$ (see Supplemental
Material SM (2022) for details). By properly tuning the overall phase factor
of $u_{1\boldsymbol{k}}$, we can map it to a real-valued 3D unit vector
$\boldsymbol{n}({\boldsymbol{k}})$, this vector is then compared with the
real-valued eigenstates vectors obtained from the tight-binding model.

Figure 4: (a) Illustration of the pump-probe technique for the measurement of
the acoustic Bloch wave functions. (b) The measured vector field formed by the
Bloch wave functions of the first acoustic bulk band which shows the meron
pattern fully consistent with Fig. 2(a). Green arrows indicate $k_{1}$ and
$k_{2}$. (c)-(d) For the vector field, the variation of the azimuth and
elevation angles along the (c) $M$-$\Gamma$-$M$ and (d)
$M$-$K$-$\Gamma$-$K^{\prime}$-$M$ lines. Results from both tight-binding model
(TBM) and experiments (Exp.) are displayed.

In Fig. 4(b), we present the distribution of the measured eigenvector
$\boldsymbol{n}({\boldsymbol{k}})$ in the whole Brillouin zone. The results
here show excellent agreement with the results in Fig. 2(a) from the tight-
binding model. We check quantitatively the azimuth and the elevation angles of
the measured eigenvector and the calculated eigenvector along two special
lines, the $M$-$\Gamma$-$M$ line [Fig. 4(c)] and the
$M$-$K$-$\Gamma$-$K^{\prime}$-$M$ line [Fig. 4(d)]. The consistency between
the experiments and theory confirm the winding of the eigenstates vector and
the meron pattern in the acoustic Bloch wave functions for the topological
Euler insulator phase.

Summary and outlook.— Our experiments unequivocally demonstrate for the first
time a gapped topological Euler system in an acoustic setup. Given the
steadily rise of such multi-gap topology in both experimental as well as
theoretical settings, our findings show that acoustic metamaterials provide
for an excellent platform to reveal the underlying physics. Indeed, the
presented meron formulation can even be generalized to, e.g., skyrmion winding
numbers of double vectors in systems with more bands Bouhon and Slager (2022).
From an experimental point of view, the observation of the pattern of the
acoustic Bloch wave functions in the Brillouin zone unveils an unprecedented
topological structure of Bloch wave functions and a salient feature of
topological Euler phases. The discoveries here may inspire future exploration
of non-Abelian topological phases with unconventional wave function geometry
and thus opens a new realm for topological physics and beyond.

Acknowledgements.— B. J., S. Q. W, Z. L. K, Z. K. L. thank the support from
National Natural Science Foundation of China (Grant Nos. 12125504 and
12074281). The work at Soochow University is also supported by Priority
Academic Program Development (PAPD) of Jiangsu Higher Education Institutions
and the Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu
Province. A. B. has been partly funded by a Marie-Curie fellowship, grant no.
101025315. R. J. S acknowledges funding from a New Investigator Award, EPSRC
grant EP/W00187X/1, as well as Trinity college, Cambridge.

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##  Appendix A The kagome tight-binding model

We here elaborate more on the proposed model in the main text. We show that
the kagome model is readily obtained from an underlying 2D hexagonal lattice
of maximal symmetry. The Bravais lattice vectors of the two-dimensional
primitive hexagonal lattice are chosen

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

with the orthonormal Cartesian frame $\langle\hat{x},\hat{y},\hat{z}\rangle$,
e.g., $\hat{x}=(1,0,0)$, $\hat{y}=(0,1,0)$, $\hat{z}=(0,0,1)$. Since we do not
consider translations in the direction perpendicular to the basal plane
($\hat{z}$), the primitive vector $\boldsymbol{a}_{3}$ vector merely there to
close the formula of the primitive reciprocal lattice vectors, that are given
by

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

Assuming that the system has the symmetry of the layer group $\\#80$, i.e.
$p6/mmm$ ($LG80$), with the point group $D_{6h}=6/mmm$, the Kagome lattice is
then readily defined from the Wyckoff position (WP) $3c$ of $LG80$. We label
$\\{A,B,C\\}$ the sub-lattice sites of WP$3c$ (i.e. these are three non-
equivalent atomic sites in the unit cell), to which attribute the positions
(see Fig. 5)

$\boldsymbol{r}_{A}=\boldsymbol{a}_{1}/2,\;\boldsymbol{r}_{B}=\boldsymbol{a}_{2}/2,\;\boldsymbol{r}_{C}=-(\boldsymbol{a}_{1}+\boldsymbol{a}_{2})/2\;.$
(3)

In the following we assume that a single s-wave orbital is occupied on each
sub-lattice site, and we define the site-orbital-Bloch basis by

$|\varphi_{\alpha},\boldsymbol{k}\rangle=\sum\limits_{\boldsymbol{R}_{n}}e^{\text{i}\boldsymbol{k}\cdot(\boldsymbol{R}_{n}+\boldsymbol{r}_{\alpha})}|w_{\alpha},\boldsymbol{R}_{n}+\boldsymbol{r}_{\alpha}\rangle\,,$
(4)

for $\alpha=A,B,C$, where the physical degrees-of-freedom are represented by
localized Wannier functions
$\langle\hat{\boldsymbol{x}}|w_{\alpha},\boldsymbol{R}_{n}+\boldsymbol{r}_{\alpha}\rangle=w_{\alpha}(\boldsymbol{R}_{n}+\boldsymbol{r}_{\alpha}-\boldsymbol{x})$.
The Bloch Hamiltonian then takes the form

$\displaystyle\mathcal{H}$
$\displaystyle=\sum\limits_{\boldsymbol{k}}\left[|\varphi_{A},\boldsymbol{k}\rangle~{}|\varphi_{B},\boldsymbol{k}\rangle~{}|\varphi_{C},\boldsymbol{k}\rangle\right]\cdot
H(\boldsymbol{k})\cdot\left[\begin{array}[]{c}\langle\varphi_{A},\boldsymbol{k}|\\\
\langle\varphi_{B},\boldsymbol{k}|\\\
\langle\varphi_{C},\boldsymbol{k}|\end{array}\right]\,,$ (5)
$\displaystyle=\sum\limits_{\boldsymbol{k}}|\boldsymbol{\varphi},\boldsymbol{k}\rangle
H(\boldsymbol{k})\langle\boldsymbol{\varphi},\boldsymbol{k}|\,.$

Allowing the nearest-neighbor (NN), next-to-nearest-neighbor (NNN), and next-
to-next-to-nearest-neighbor (NNNN) hopping processes, with their corresponding
tunnelling amplitudes written $t$, $t^{\prime}$, and $t^{\prime\prime}$,
respectively (see also Fig. 5), we obtain the minimal kagome model tight-
binding model

$H(\boldsymbol{k})=\left(\begin{array}[]{ccc}\varepsilon_{A}+t^{\prime\prime}c_{1}^{\prime\prime}(\boldsymbol{k})&tc_{2}(\boldsymbol{k})+t^{\prime}c^{\prime}_{2}(\boldsymbol{k})&tc_{1}(\boldsymbol{k})+t^{\prime}c^{\prime}_{1}(\boldsymbol{k})\\\
tc_{2}(\boldsymbol{k})+t^{\prime}c^{\prime}_{2}(\boldsymbol{k})&\varepsilon_{B}+t^{\prime\prime}c_{2}^{\prime\prime}(k)&tc_{3}(\boldsymbol{k})+t^{\prime}c^{\prime}_{3}(\boldsymbol{k})\\\
tc_{1}(\boldsymbol{k})+t^{\prime}c^{\prime}_{1}(\boldsymbol{k})&tc_{3}(\boldsymbol{k})+t^{\prime}c^{\prime}_{3}(\boldsymbol{k})&\varepsilon_{C}+t^{\prime\prime}c_{3}^{\prime\prime}(\boldsymbol{k})\end{array}\right),$
(6)

where the finite Fourier terms are defined, using
$c[\boldsymbol{r}](\boldsymbol{k})=\cos(\boldsymbol{k}\cdot\boldsymbol{r})$,
by

$c_{i}(\boldsymbol{k})=c[\boldsymbol{\delta}_{i}](\boldsymbol{k}),\;c^{\prime}_{i}(\boldsymbol{k})=c[\boldsymbol{\delta}^{\prime}_{i}](\boldsymbol{k}),\;c^{\prime\prime}_{i}(\boldsymbol{k})=c[\boldsymbol{\delta}^{\prime\prime}_{i}](\boldsymbol{k}),$

for $i=1,2,3$, with the NN bond vectors

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

the NNN bond vectors,

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

and finally the NNNN bond vectors,

$\boldsymbol{\delta}^{\prime\prime}_{1}=\boldsymbol{a}_{1},\;\boldsymbol{\delta}^{\prime\prime}_{2}=\boldsymbol{a}_{2},\;\boldsymbol{\delta}^{\prime\prime}_{3}=-\boldsymbol{a}_{1}-\boldsymbol{a}_{2}.$

Figure 5: Kagome lattice structure. Kagome lattice (black dots). Red lines are
the first-neighbor bonds of the kagome lattice with a tunneling amplitude
written $t$. The blue (yellow) lines are the second (third) neighbor bonds
with $t^{\prime}$, and $t^{\prime\prime}$, respectively.

Let us write the momentum in units of the reciprocal lattice vectors as
$\boldsymbol{k}=k_{1}\boldsymbol{b}_{1}/\pi+k_{2}\boldsymbol{b}_{2}/\pi$, and
taking $(k_{1},k_{2})\in[-\pi/2,\pi/2]^{2}$ the domain of the Brillouin zone.
The first-neighbor terms thus are

$\displaystyle c_{1}(\boldsymbol{k})$
$\displaystyle=\cos[k_{2}],\;\quad\quad\quad
c_{3}(\boldsymbol{k})=\cos[k_{1}],$ (7) $\displaystyle c_{2}(\boldsymbol{k})$
$\displaystyle=\cos[k_{1}+k_{2}].$

The second-neighbor terms are

$\displaystyle c^{\prime}_{1}(\boldsymbol{k})$
$\displaystyle=\cos[2k_{1}+k_{2}],\;\quad
c^{\prime}_{3}(\boldsymbol{k})=\cos[k_{1}+2k_{2}],$ (8) $\displaystyle
c^{\prime}_{2}(\boldsymbol{k})$ $\displaystyle=\cos[k_{1}-k_{2}],$

and the third-neighbor terms are

$c^{\prime\prime}_{1}(\boldsymbol{k})=\cos[2k_{1}],\;c^{\prime\prime}_{2}(\boldsymbol{k})=\cos[2k_{2}],\;c^{\prime\prime}_{3}(\boldsymbol{k})=\cos[2(k_{1}+k_{2})].$
(9)

###  A.1 Reality condition from $C_{2}{\cal T}$ (${\cal PT}$)

The $C_{2}{\cal T}$ (or equivalently ${\cal PT}$) symmetry (with $[C_{2}{\cal
T}]^{2}=[{\cal PT}]^{2}=+1$) guaranties the existence of a choice of a Bloch-
orbital basis in which the Bloch Hamiltonian matrix $H(\boldsymbol{k})$ is
real Bouhon  _et al._ (2020a, b). Since $C_{2z}{\cal T}$ acts the Bloch site
basis as

${}^{C_{2}{\cal
T}}|\boldsymbol{\varphi},\boldsymbol{k}\rangle=|\boldsymbol{\varphi},m_{z}\boldsymbol{k}\rangle\cdot\mathbb{1}_{3}\mathcal{K}\,,$
(10)

where $\mathbb{1}_{3}$ is the $3\times 3$ identity matrix, and $\mathcal{K}$
is complex conjugation, the $C_{2}{\cal T}$ symmetry leads to the following
constraint on the Bloch matrix
$H(\sigma_{h}\boldsymbol{k})^{*}=H(\boldsymbol{k})$ ($\sigma_{h}$ is the basal
mirror symmetry), i.e. it must be real since
$m_{z}\boldsymbol{k}=\boldsymbol{k}$ in the 2D Brillouin zone.

###  A.2 Spectral decomposition

The eigenvalue problem of the system is defined by the equation

$\mathcal{H}|\psi_{n},\boldsymbol{k}\rangle=E_{n}(\boldsymbol{k})|\psi_{n},\boldsymbol{k}\rangle,$
(11)

where $|\psi_{n},\boldsymbol{k}\rangle$ is the Bloch eigenstate of the $n-th$
eigenenergy $E_{n}(\boldsymbol{k})$. In the following we always assume that
the eigenvalues and the corresponding eigenvectors are ordered in energy from
the bottom, i.e. $E_{1}(\boldsymbol{k})\leq E_{2}(\boldsymbol{k})\leq
E_{3}(\boldsymbol{k})$. Decomposing the Bloch eigenstates in terms of the
Bloch-orbital states,

$|\psi_{n},\boldsymbol{k}\rangle=\sum\limits_{\alpha=A,B,C}|\varphi_{\alpha},\boldsymbol{k}\rangle[R(\boldsymbol{k})]_{\alpha
n}\,,$ (12)

the eigenvalue problem reduces to

$\displaystyle H_{\alpha\beta}(\boldsymbol{k})[u_{n}(\boldsymbol{k})]_{\beta}$
$\displaystyle=E_{n}(\boldsymbol{k})\,[u_{n}(\boldsymbol{k})]_{\alpha}\,,$
(13)

where $\\{u_{n}(\boldsymbol{k})\\}_{n=1,2,3}$ are real orthogonal Bloch
eigenvectors. The spectral decomposition of the Bloch matrix
$H(\boldsymbol{k})$ is then

$\displaystyle H(\boldsymbol{k})$
$\displaystyle=R(\boldsymbol{k})\cdot\left[\begin{array}[]{ccc}E_{1}(\boldsymbol{k})&0&0\\\
0&E_{2}(\boldsymbol{k})&0\\\ 0&0&E_{3}(\boldsymbol{k})\end{array}\right]\cdot
R(\boldsymbol{k})^{T}\,,$ (14) $\displaystyle R(\boldsymbol{k})$
$\displaystyle=\left[u_{1}(\boldsymbol{k})~{}u_{2}(\boldsymbol{k})~{}u_{3}(\boldsymbol{k})\right]\in\mathsf{O}(3)\,,$

where the diagonalizing matrix $[R_{\alpha
n}=u_{n,\alpha}]_{\alpha=A,B,C;n=1,2,3}$ is composed of the column Bloch
eigenvectors and is orthogonal since the matrix Hamiltonian
$H(\boldsymbol{k})$ is real.

###  A.3 “Orbital” Zak phases and “flag atomic limit”

We readily note that the configuration of the sub-lattice sites, in which they
are shifted from the center of the unit cell and are not related via any
Bravais lattice vector (this is why we call them “inequivalent”), implies the
following nontrivial transformation of the Bloch matrix under a reciprocal
lattice translation $\boldsymbol{K}=m\boldsymbol{b}_{1}+n\boldsymbol{b}_{2}$
($m,n\in\mathbb{Z}$),

$V(\boldsymbol{K})\cdot H(\boldsymbol{k}+\boldsymbol{K})\cdot
V(\boldsymbol{K})^{\dagger}=H(\boldsymbol{k})\,,$ (15)

with
$V(\boldsymbol{K})=\mathrm{diag}[e^{\text{i}\boldsymbol{K}\cdot\boldsymbol{r}_{A}},e^{\text{i}\boldsymbol{K}\cdot\boldsymbol{r}_{B}},e^{\text{i}\boldsymbol{K}\cdot\boldsymbol{r}_{C}}]$,
which follows directly from the definition of the Bloch basis Eq. (4). Then,
imposing the translational symmetry of the Bloch eigenstates, i.e.

$|\psi_{n},\boldsymbol{k}+\boldsymbol{K}\rangle=|\psi_{n},\boldsymbol{k}\rangle\,,$
(16)

we get

$u_{n}(\boldsymbol{k}+\boldsymbol{K})=V(\boldsymbol{K})^{T}\cdot
u_{n}(\boldsymbol{k})s_{n}\,,$ (17)

where $s_{n}=\pm 1$ is a free gauge sign. The convention is to use the
periodic gauge, i.e. setting $s_{n}=1$.

It is instructive to consider a trivial limiting atomic phase of the kagome
model where there is no hopping and all the bands are gapped by setting
different onsite energies. (Note that this implies the breaking of the
hexagonal symmetry of the system, see below.) Setting
$\varepsilon_{A}=-\varepsilon_{B}=1$, $\varepsilon_{B}=0$, and
$t=t^{\prime}=t^{\prime\prime}=0$, we have

$\displaystyle E_{1}=\varepsilon_{C}=-1,~{}E_{2}$
$\displaystyle=\varepsilon_{B}=0,~{}E_{3}=\varepsilon_{A}=1\,,$ (18)
$\displaystyle[u_{n}]_{\alpha}$ $\displaystyle=\delta_{n\alpha}\,.$

We can then readily derive the Berry phase (Zak phases) per band integrated
along $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$, noting that each band has
a pure orbital composition ($\mathcal{B}_{1}\equiv\mathcal{B}_{C}$,
$\mathcal{B}_{2}\equiv\mathcal{B}_{B}$,
$\mathcal{B}_{3}\equiv\mathcal{B}_{A}$). Writing
$\gamma^{(\mathcal{B}_{n})}_{B,i}=\gamma^{(\mathcal{B}_{n})}_{B}[\Gamma+\boldsymbol{b}_{i}\leftarrow\Gamma]$
with $i=1,2$, we have

$e^{-\text{i}\gamma^{(\mathcal{B}_{n})}_{B,i}}=e^{\text{i}\boldsymbol{b}_{i}\cdot\boldsymbol{r}_{n}}\,,$
(19)

and writing the vector
$\boldsymbol{\gamma}^{(\mathcal{B}_{n})}_{B}=(\gamma^{(\mathcal{B}_{n})}_{B,1},\gamma^{(\mathcal{B}_{n})}_{B,2})$,
we find Jiang  _et al._ (2021)

$\left\\{\boldsymbol{\gamma}^{(\mathcal{B}_{1})}_{B}=(\pi,\pi),~{}\boldsymbol{\gamma}^{(\mathcal{B}_{2})}_{B}=(0,\pi),~{}\boldsymbol{\gamma}^{(\mathcal{B}_{3})}_{B}=(\pi,0)\right\\}~{}\text{mod}\,2\pi\,.$
(20)

In this work we call these non-zero Berry phases the orbital $\pi$-Zak phases,
since these directly originate from the atomic off-center configuration of the
orbitals in the unit cell. Indeed, we remind that the Zak phases can be
readily interpreted as the center of localized “band” Wannier functions Zak
(1989), here matching with the pure orbital degrees of freedom. Hence, we can
readily verify that the $\pi$-Zak phases are in direct correspondence with the
off-center locations of the orbital in the Wyckoff position $3c$ of $LG80$.

Interestingly, this completely “trivial” phase from the side of the band
structure (i.e. $H(\boldsymbol{k})$ is diagonal and has no momentum
dependence), it is nevertheless non-orientable, i.e. it exhibits $\pi$-Berry
phases along one or several non-contractible paths of the Brillouin zone
torus. Such $\pi$-Berry phases imply that associated Dirac strings must cross
the Brillouin zone leading to the non-trivial interaction between these and
the nodal points of adjacent gaps. For a detailed discussion see Ref. Jiang
_et al._ (2021); Bouhon  _et al._ (2020b); Peng  _et al._ (2022a). We
furthermore conclude that the above phase is an atomic limit without
obstruction (i.e. without shift of the center of the “band” Wannier orbitals
as compared to the atomic orbitals) Bradlyn  _et al._ (2017). We call this
atomic limit of the kagome lattice, the flag atomic limit, where “flag” refers
to the fact that all the bands are gapped, as opposed to the Grassmannian
which is characterized by a single gap Bouhon  _et al._ (2020b).

In Section D, we discuss the intricate effect of the cancellation of the
orbital Zak phases due to the presence of band Zak phases that now originate
from the nontrivial winding of the Bloch eigenvectors along non-contractible
paths of the Brillouin zone. Anticipating the discussion, we find that this
cancellation of Zak phases converts the non-orientable flag atomic band limit
into an orientable Euler phase with a meron vector field structure (instead of
a full Skyrmion).

##  Appendix B Fragile crystalline topology

When $\varepsilon_{A}=\varepsilon_{B}=\varepsilon_{C}$, the tight-binding
model has the symmetries of the hexagonal layer group LG80, with the point
symmetry group $D_{6h}$. One important remark is in place here. The lattice of
Fig. 5, on which the tight-binding is based, is a slight idealization of the
real acoustic system on which we make the measurements. Indeed, the details of
the acoustic structure (see Fig. 3(a) of the main text) effectively breaks
$C_{2z}$ and the basal mirror symmetry $\sigma_{h}$, while inversion $I$
symmetry is preserved. The lowered symmetry of the printed system is given by
the layer group $LG71$, with the point group $D_{3d}$. However, we find that
the band structure of the Bloch wave functions of the acoustic wave equation
for the printed structure is completely insensitive to this lowered symmetry.
Furthermore, at the tight-binding level, there is no way to break $C_{2}$
(while preserving $I$) without doubling the number of sites per unit cell,
i.e. converting the Wyckoff position $3c$ of $LG80$ to the Wyckoff position
$6f$ of the layer group $LG71$. In the whole work we thus use the full
$D_{6h}$ symmetry, since it is the relevant one to characterize the band
structure.

Figure 6: Split elementary band representation (EBR) of the gapped kagome
system. We give the IRREPs for the effective higher symmetry group of the
system ($D_{6h}$) and, in brackets, those for the reduced symmetry group of
the material realization ($D_{3d}$) (we use the convention of the Bilbao
Crystallographic Server Aroyo  _et al._ (2006)). Below each high-symmetry
points we have noted the corresponding little co-group.

###  B.1 Split EBR and symmetry-indicated winding of the two-band Wilson loop

We show in Fig. 6 the band structure of the gapped kagome model,
$(t,t^{\prime},t^{\prime\prime})=(-1,-1,-0.8)$, along the high-symmetry lines,
with the IRREPs of the symmetry group $D_{6h}$. The set of IRREPs of Fig. 6
induced by the Wyckoff position $3c$ of $LG80$ constitute an elementary band
representation (EBR) Zak (1980, 1981); Michel and Zak (1999, 2000, 2001);
Bradlyn  _et al._ (2017); Höller and Alexandradinata (2018). From of the
gaping of the EBR, and since the IRREPs of the two-band subspace above the gap
are not compatible with any band representation of $LG80$ Aroyo  _et al._
(2006), we deduce that the gapped kagome system must host a nontrivial
crystalline topology, because the two-band subspace cannot be mapped to
symmetric and localized Wannier functions Bradlyn  _et al._ (2017). However,
given the fact that the lower single band has no stable topology (nontrivial
Euler and second Stiefel-Whitney classes require a minimum rank of 2, see
below), we conclude that the system must host a fragile topology Po  _et al._
(2018); Bouhon  _et al._ (2019).

We have shown in Ref. Bouhon  _et al._ (2019) that the nontrivial topology of
such a split EBR can be captured through the symmetry-constrained winding of
(two-band) Wilson loop. Considering one patch bounded by the non-contractible
paths $l_{\Gamma K}:\Gamma\rightarrow
K\rightarrow\Gamma+\boldsymbol{b}_{1}-\boldsymbol{b}_{2}$ and $l_{\Gamma
M}:\Gamma\rightarrow
M\rightarrow\Gamma+\boldsymbol{b}_{1}-\boldsymbol{b}_{2}$, [see Fig. 7(a)],
the flow of Wilson loop upon scanning the patch by smoothly deforming
$l_{\Gamma K}$ to $l_{\Gamma M}$ must be quantized. This results from the
symmetry quantization of the Wilson loop phases to $[2\pi/3,-2\pi/3]$ over
$l_{\Gamma K}$, and $[\pi,-\pi]$ over $l_{\Gamma M}$, see Bouhon  _et al._
(2019) for a detailed derivation. We show in Fig. 7(b) the computed Wilson
loop over the patch, obtained by deforming $l_{\Gamma K}$ (at $t=0$) to
$l_{\Gamma M}$ (at $t=1$). Since the patch covers one sixth of the Brillouin
zone, we then obtained that the two-band Wilson loop winds one time in total,
i.e. each Wilson loop branch winds by $2\pi$, see Fig. 7(c).

We emphasize that while the IRREPs can be used to identity a fragile
crystalline topological phase coming from the splitting of an EBR, we show
below that the underlying Euler topology is preserved upon the breaking of the
hexagonal symmetries, while preserving $C_{2}{\cal T}$ (or equivalently ${\cal
PT}$), even though the Wilson loop is not quantized by the unitary crystalline
symmetries anymore, see Fig. 7(d) and (e). In that sense, the Euler topology
is more fundamental, similarly to the Chern topology in the complex case, and
thus goes beyond the symmetry indicated topological phases Kruthoff  _et al._
(2017); Bouhon and Black-Schaffer (2017); Po  _et al._ (2017); Bradlyn  _et
al._ (2017).

(a)
---

(b) | (c)
---|---
|
(d) | (e)
|

Figure 7: Quantized Wilson loop indicating a fragile topological crystalline
phase. (a) Symmetric patch of the Brillouin zone (in blue) bounded by
$l_{\Gamma K}$ and $l_{\Gamma M}$ over which we consider the flow of Wilson
loop, by smoothly deforming $l_{\Gamma K}$ to $l_{\Gamma M}$. Considering the
split ERB of Fig. 6, the panel (b) shows the quantization of the winding of
the two-band Wilson loop over the patch from $l_{\Gamma K}$ at $t=0$ to
$l_{\Gamma M}$ at $t=1$ (see Bouhon  _et al._ (2019) for a detailed
derivation). The panel (c) shows the resulting complete winding of the Wilson
loop over the whole Brillouin zone, i.e. $6|\pi-2\pi/3|=2\pi$, given that the
patch covers one sixth of the Brillouin zone. (c) and (e) are the same as (b)
and (c) but now with broken crystalline symmetries, while preserving
$C_{2}{\cal T}$ ($IT$).

##  Appendix C The “real” topology

Given that the system satisfies time reversal symmetry ($T$) it belongs to the
class $\mathsf{AI}$ of the tenfold-way of the periodic table of topological
phases of matter Schnyder  _et al._ (2008); Kitaev (2009). In particular, the
system has $C_{2z}T$ and $IT$ symmetries that both square to the identity
($[C_{2z}T]^{2}=[IT]^{2}=+1$), with $T$ the time reversal, $C_{2z}$ the $\pi$
rotation around the $\hat{z}$ axis (perpendicular to the basal plane and
centered at the origin of the unit cell chosen at the center of the hexagons),
and $I$ the inversion symmetry (see the lattice in Fig. 5, on which the model
is based). The presence of an anti-unitary symmetry that squares to identity
and leaves the momentum invariant implies the existence of a Bloch-orbital
basis in which the matrix Hamiltonian $H(\boldsymbol{k})$ is real Bouhon  _et
al._ (2020a, b). In the case of the kagome model, this real gauge is readily
given by the $\\{A,B,C\\}$-sub-lattice site basis presented above. (In generic
cases, assuming a hermitian Hamiltonian, the real basis is obtained via the
Takagi factorization of symmetry unitary matrices, see the appendix of Chen
_et al._ (2022) where we present a derivation based on the singular value
decomposition.) The reality condition implies that the topology of the band
structure fundamentally corresponds to the topology of real vector bundles
Hatcher (2003). The $\mathbb{Z}_{2}$ quantization of the Berry phase in the
case of a real Hamiltonian directly relates to the first Stiefel-Whitney class
of real vector bundles Hatcher (2003); Ahn  _et al._ (2018, 2019). This one-
dimensional invariant capture the orientability of the vector bundle. Then,
orientable real vector bundles of rank 2, i.e. with trivial first Stiefel-
Whitney class, are classified by the Euler class $\chi\in\mathbb{Z}$. Any real
vector bundle of higher rank, orientable or non-orientable, is classified by
the reduced $\mathbb{Z}_{2}$ second Stiefel-Whitney class Ahn  _et al._
(2018); Zhao and Lu (2017); Ahn  _et al._ (2019). The refined and systematic
homotopy classification of orientable three-band and four-band real gapped
phases has been presented in Ref. Bouhon  _et al._ (2020b), with the
systematic derivation of minimal tight-binding models presented in Ref. Bouhon
and Slager (2022). We conclude this general discussion by emphasizing that the
Euler (Stiefel-Whitney) class topology constitutes the basic topology of real
gapped Hamiltonian, similarly to the fundamental Chern class in the complex
case.

As shown in Fig. 7(d), the total winding of Wilson loop, which indicates the
non-zero Euler class Bouhon  _et al._ (2019); Ahn  _et al._ (2018, 2019);
Bzdušek and Sigrist (2017); Bouhon  _et al._ (2020b), upon the breaking of the
hexagonal symmetries (and preserving $C_{2}{\cal T}$) leading to the absence
of quantization of the Wilson loop over special symmetric paths (c), shows
that the Euler topology goes beyond the framework of symmetry-indicated
topologies. See Ref. Bouhon  _et al._ (2020b) and Bouhon and Slager (2022)
with plenty of examples of models with a variety of difference Euler phases
that are not indicated by symmetry.

The topology of a real orientable $1+2$-gapped three-band system, i.e.
$E_{1}(\boldsymbol{k})<E_{2}(\boldsymbol{k})\leq E_{3}(\boldsymbol{k})$, is
classified by the Euler class Ahn  _et al._ (2018, 2019); Bouhon  _et al._
(2020a); Zhao and Lu (2017)

$\displaystyle\chi$
$\displaystyle=\dfrac{1}{2\pi}\int_{\text{BZ}}\left[\partial_{k_{x}}u_{2}\partial_{k_{y}}u_{3}-\partial_{k_{y}}u_{2}\partial_{k_{x}}u_{3}\right]dk_{x}\wedge
dk_{y}\,,$ (21)
$\displaystyle=\dfrac{1}{2\pi}\int_{\text{BZ}}u_{1}(\boldsymbol{k})\cdot\left[\partial_{k_{x}}u_{1}(\boldsymbol{k})\times\partial_{k_{y}}u_{1}(\boldsymbol{k})\right]\in(2)\mathbb{Z}\,,$

where we note the prefactor $1/(2\pi)$ in front of the Skyrmion number
(instead of the usual prefactor $1/(4\pi)$), leading to the doubling of the
Euler class whenever the vector field $u_{1}(\boldsymbol{k})$ winds fully over
the unit sphere, i.e. forming a Skyrmion vector field. In the next Section, we
discuss the winding of the vector field $u_{1}(\boldsymbol{k})$ and show that
the gapped kagome phase hosts a nontrivial Euler phase, with the caveat that
$u_{1}(\boldsymbol{k})$ winds only as meron (i.e. it covers half of the unit
sphere) instead of a full skyrmion.

##  Appendix D From non-orientable atomic flag limit to orientable meron Euler
phase

We have noted in Appendix A.3 that flag atomic limit of the kagome lattice is
non-orientable, i.e. with $\pi$-Zak phases, due to the off-centered location
of the atomic orbitals. We present in the main text the Euler topology of the
gapped kagome. In principle, however, the Euler class is not defined in non-
orientable phases (as it is not defined for non-orientable real vector
bundles, i.e. hosting nontrivial first Stiefel-Whitney class), see however
333In a sequel, we will show that the Euler class, and the winding of Wilson
loop are still relevant to characterize non-orientable phases, with the caveat
that the topological quantities in those cases do depend on the
parametrization of the integral over the Brillouin zone, and the flow of base
loops, respectively.. We here resolve this apparent contradiction.

The vector field of the first Bloch eigenvector $u_{1}(\boldsymbol{k})$ of the
gapped kagome phase, shown in Fig. 2(a) of the main text, clearly shows the
presence of $\pi$-winding across non-contractible paths of the Brillouin zone.
This is very clear for instance along the path
$[\Gamma-\boldsymbol{b}_{1}/2-\boldsymbol{b}_{2}/2\rightarrow\Gamma+\boldsymbol{b}_{1}/2-\boldsymbol{b}_{2}/2]=\\{\boldsymbol{k}|k_{2}=\pm\pi/2,\,k_{1}\in[-\pi/2,\pi/2]\\}$
connecting the three inequivalent $M$ points: in Fig. 2(a) the arrow is blue
when pointing downwards and red when pointing upwards. This follows from (see
Appendix A.3)

$\displaystyle
u_{1}\left(\Gamma+\frac{\boldsymbol{b}_{1}}{2}-\frac{\boldsymbol{b}_{2}}{2}\right)$
$\displaystyle=V(\boldsymbol{b}_{1})^{T}\cdot
u_{1}\left(\Gamma-\frac{\boldsymbol{b}_{1}}{2}-\frac{\boldsymbol{b}_{2}}{2}\right)$
(22) $\displaystyle=\mathrm{diag}[-1,-1,1]\cdot
u_{1}\left(\Gamma-\frac{\boldsymbol{b}_{1}}{2}-\frac{\boldsymbol{b}_{2}}{2}\right)\,,$
$\displaystyle\Leftrightarrow\quad(-1,0,0)^{T}$
$\displaystyle=\mathrm{diag}[-1,-1,1]\cdot(1,0,0)^{T}\,.$

We thus conclude that the $\pi$-orbital Zak phases are compensated by
$\pi$-band Zak phases, i.e. coming from the winding of the Bloch eigenvectors.
All in all, this results in an effectively orientable phase. As a consequence,
the Euler class is naturally well defined. One very interesting consequence,
is that the Euler class of the gapped kagome phase is one, i.e. it is odd.
This must be contrasted with the constraint that every three-band system that
hosts an orientable flag atomic limit only allows nontrivial Euler phases with
an even Euler class (thus the factor $(2)$ in front of $\mathbb{Z}$ in Eq.
(21)).

Let us elaborate further. The flattened Hamiltonian of any $1+2$-three-band
system is (taking the eigenvalues to be $E_{1}=-E_{2}=E_{3}=1$) Bzdušek and
Sigrist (2017); Bouhon  _et al._ (2020a)

$H(\boldsymbol{k})\rightarrow Q(\boldsymbol{k})=2u_{1}(\boldsymbol{k})\cdot
u_{1}(\boldsymbol{k})^{T}-\mathbb{1}_{3}\;.$ (23)

Because of the quadratic dependence on $u_{1}$, whenever the vector $u_{1}$
winds fully over the unit sphere an number $N$ of times, the Euler class is
$2N$, via Eq. (21). In the case of the gapped kagome phase, we find that the
vector $u_{1}$ only winds over the half of the unit spheres. This can be
realted to the fact that the vector field is not periodic over a single
Brillouin zone, see Fig. 2(a) in the main text. However, the integrand
$u_{1}\cdot(\partial_{x}u_{1}\times\partial_{y}u_{1})$ in Eq. (21) is
periodic, leading to an integer value whenever $u_{1}$ covers one half sphere
since then the integral compute the half-solid angle $2\pi$.

We can now conclude that there are two classes of three-band systems with
Euler topology. On one hand, the systems that host an orientable flag atomic
limit give rise to Euler phases with an even Euler class. We call these the
Skyrmionic Euler phases. On the other hand, the systems that host a non-
orientable flag atomic limit, give rise to Euler phases with an odd Euler
class. We call these the meronic Euler phases.