# Topological continuum charges of acoustic phonons in 2D

Gunnar F. Lange<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University
of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Adrien
Bouhon<EMAIL_ADDRESS>Nordic Institute for Theoretical Physics
(NORDITA), Stockholm, Sweden  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 21 Uppsala, Sweden  Bartomeu Monserrat
<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of Cambridge, J.J.
Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Department of Materials
Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road,
Cambridge CB3 0FS, United Kingdom  Robert-Jan Slager<EMAIL_ADDRESS>TCM
Group, Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom

###### Abstract

We analyze the band topology of acoustic phonons in 2D materials by
considering the interplay of spatial and internal symmetries with additional
constraints that arise from the physical context. These supplemental
constraints trace back to the Nambu-Goldstone theorem and the requirements of
structural stability. We show that this interplay can give rise to previously
unaddressed non-trivial nodal charges that are associated with the crossing of
the acoustic phonon branches at the center ($\Gamma$-point) of the phononic
Brillouin zone. We moreover apply our perspective to the concrete context of
graphene, where we demonstrate that the phonon spectrum harbors these kinds of
non-trivial nodal charges. Apart from its fundamental appeal, this analysis is
physically consequential and dictates how the phonon dispersion is affected
when graphene is grown on a substrate. Given the generality of our framework,
we anticipate that our strategy that thrives on combining physical context
with insights from topology should be widely applicable in characterizing
systems beyond electronic band theory.

##  I Introduction

The interplay between symmetry and topology has been well studied in
electronic band structures for a long time, culminating in classification
schemes that predict topology based only on the space group and the internal
symmetries of the system Fu (2011); Shiozaki and Sato (2014); Höller and
Alexandradinata (2018); Juričić  _et al._ (2012); Chiu  _et al._ (2016); Song
_et al._ (2018); Po  _et al._ (2017); Bradlyn  _et al._ (2017); Slager  _et
al._ (2012); Kruthoff  _et al._ (2017); Bouhon  _et al._ (2020a, 2021);
Watanabe  _et al._ (2018); Slager (2019); Elcoro  _et al._ (2020); Po  _et
al._ (2018); Lange  _et al._ (2021); Borgnia  _et al._ (2020); Song  _et al._
(2020); Peri  _et al._ (2020). The same machinery has also recently been
applied to phononic systems Mañes (2020); Peng  _et al._ (2020); Tang and Cao
(2021); Li  _et al._ (2021), where the Bloch Hamiltonian of electrons is
replaced with the dynamical matrix of phonons. The band topology of phononic
systems is then described using the spinless space groups, that is, the
phonons are modelled using the symmetries of spinless electrons. As the
dynamical matrix naturally includes time-reversal symmetry (TRS), this
corresponds to the Altland-Zirnbauer (AZ) Altland and Zirnbauer (1997);
Schnyder  _et al._ (2008); Kitaev (2009) class AI.

However, phonons are not just spinless electrons. Whilst AZ class AI (possibly
augmented by spatial symmetries) correctly captures the symmetry content of
phonons, there are additional physical properties that set phonons apart from
electrons. The most relevant of these Kane and Lubensky (2013); Po  _et al._
(2016); Liu  _et al._ (2020) are:

  * •

Phonon frequencies of stable structures are non-negative, so that the
dynamical matrix is positive semidefinite

  * •

Phonons (being bosons) do not couple directly to magnetic fields, so TRS is
not easily broken (see, however, Appendix A.2)

  * •

Phonons satisfy the acoustic sum rule, e.g. they support long-wavelength
excitations with vanishing frequency. These arise as a consequence of the
Nambu-Goldstone (NG) theorem Nambu (1960); Goldstone (1961); Arraut (2017);
Watanabe (2020); Else (2021).

We will refer to these as additional physical constraints. Earlier work on
phononic topology Prodan and Prodan (2009); Zhang  _et al._ (2010); Li  _et
al._ (2012); Kane and Lubensky (2013); Maldovan (2013); Roman and Sebastian
(2015); Wang  _et al._ (2015a, b); Yang  _et al._ (2015); Nash  _et al._
(2015); Peano  _et al._ (2015); Huber (2016); Süsstrunk and Huber (2016);
Kariyado and Slager (2019, 2021); Po  _et al._ (2016); Liu  _et al._ (2017,
2020) usually incorporates these constraints by moving away from a direct
dynamical matrix formulation. One strategy, introduced in Ref. Kane and
Lubensky (2013), is to map the bosonic phonon problem to a fermionic problem
Po  _et al._ (2016) by considering the square root of the dynamical matrix.
This replaces the positive definiteness condition with a particle-hole
symmetry, leading to AZ class BDI, and also gives a natural way to include TRS
breaking Liu  _et al._ (2017, 2020).

Here, by contrast, we deal directly with the dynamical matrix, and discuss how
the additional physical constraints interplay with the conventional symmetry
analysis. Concretely, we study the nodal charge of acoustic phonons in a 2D
material at the $\Gamma$ point [$\boldsymbol{q}=(0,0)$] of the Brillouin zone.
Allowing the material to flex out-of-plane, the NG theorem Nambu (1960);
Goldstone (1961); Arraut (2017); Watanabe (2020); Else (2021) predicts that
three acoustic bands will be degenerate at $\Gamma$, forming a triple point.
We assume the presence of spinless TRS $\mathcal{T}$ throughout (this is
discussed in Appendix A.2), e.g. $\mathcal{T}^{2}=+1$ so that we are in AZ
class AI. As a result, the spatial symmetries of our system are described by
the 80 layer groups with spinless TRS and (Editor)(2010) (Editor).

However, none of these layer groups have a three-dimensional irreducible
representation (IRREP) Bradley and Cracknell (1972); Aroyo  _et al._ (2006),
so that triply degenerate points are not stabilized by internal or spatial
symmetries in AZ class AI in 2D. Such triple points are therefore not
anticipated from a pure symmetry analysis, and arise from the NG theorem.
Imposing such a triple point, we can then use the machinery of homotopy theory
to compute the nodal charge of the triple point. This is computation is
simplified if the system has a unitary symmetry $\mathcal{P}$ taking
$\boldsymbol{q}\rightarrow-\boldsymbol{q}$ and satisfying
$(\mathcal{P}\mathcal{T})^{2}=+1$, because this allows us to restrict to real
topology Shiozaki and Sato (2014); Bouhon  _et al._ (2020b); Chen  _et al._
(2021); Tiwari and Bzdušek (2020); Bouhon  _et al._ (2020a); Bzdusek and
Sigrist (2017); Bouhon  _et al._ (2019); Wu  _et al._ (2019); Ünal  _et al._
(2020), as discussed in Sec. II.2. We will refer to $\mathcal{P}$ as a
generalized inversion symmetry. Such a symmetry does not necessarily exist
globally in phononic systems, but we show in Appendix A that the physical
constraints above force such a symmetry to exist close to the $\Gamma$-point.
In 2D, $\mathcal{P}$ is given by a twofold rotation.

We find that, with this additional symmetry, there is a nodal charge
associated with the acoustic phonons in 2D. However, this charge is only
associated with two of the bands, the third band being degenerate only by
virtue of the NG theorem. This explains why, in 2D materials, one of the
acoustic bands can gap out when the material is grown on a substrate, as
confirmed experimentally for graphene Aizawa  _et al._ (1990); Al Taleb and
Farías (2016). The substrate allows for a violation of the NG theorem,
splitting off one of the bands, whereas the other two bands are stabilized by
the nodal charge. This is discussed in detail in Sec. V. Due the generality of
our approach, we emphasize that graphene is nonetheless just a specific
example of this universal perspective. We note that a similar analysis was
recently carried out in 3D in Ref. Park  _et al._ (2021), and we comment on
the connection of their results to ours throughout.

This paper is structured as follows: In Sec. II we introduce the model for 2D
acoustic phonons, and discuss some general symmetry considerations. In Sec.
III, we discuss the possible topology, and apply it to the 2D system in Sec.
IV. We then exemplify these concepts by applying the machinery to graphene in
Sec. V, discussing how a substrate modifies the phonon dispersion. We conclude
in Sec. VI.

##  II Continuum models from elasticity theory

###  II.1 Flexural phonons in 2D

We begin by introducing a continuum model for acoustic phonons based on
classical elasticity theory in 2D Landau  _et al._ (1986). The analogous model
in 3D was studied in Park  _et al._ (2021), and we include it for completeness
in Appendix B.1 where we also discuss the model in 1D and show that it is
trivial.

Classical continuum theory describes a 2D material as an elastic membrane in
the $xy$-plane which can flex in the $z$-plane, giving rise to flexural modes
Jiang  _et al._ (2015). The Lagrangian density for such a system is written in
terms of the in-plane displacement field
$\boldsymbol{u}(x,y)=(u_{1}(x,y),u_{2}(x,y))$ and the out-of plane
displacement field $h(x,y)$. These fields are defined respectively as the in-
plane and out of plane deviations of the atoms from their equilibrium
position. Explicitly, the Lagrangian density for a flexible membrane is given
by Mariani and Von Oppen (2008); Sachdev and Nelson (1984); Landau  _et al._
(1986):

$\mathcal{L}=\frac{\rho_{0}}{2}(\mathbf{\dot{u}}^{2}+\dot{h}^{2})-\frac{1}{2}\kappa_{0}(\nabla^{2}h)^{2}-\mu\overset{\text{\scriptsize$\leftrightarrow$}}{u}_{ij}^{2}-\frac{1}{2}\lambda\overset{\text{\scriptsize$\leftrightarrow$}}{u}_{kk}^{2},$
(1)

Where $\mu$ and $\lambda$ are Lamé parameters and $\rho_{0}$ and $\kappa_{0}$
are the stiffness in- and out-of plane, respectively. The strain tensor
$\overset{\text{\scriptsize$\leftrightarrow$}}{u}_{ij}$ is defined as:

$\overset{\text{\scriptsize$\leftrightarrow$}}{u}_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i}+\partial_{i}h\partial_{j}h)$
(2)

Expanding Eq. (1) to quadratic order in the displacements defines what we will
refer to as the harmonic approximation. This is valid whenever phonon-phonon
interactions are negligible, which we assume throughout (for a discussion of
such terms, see Ref. Sachdev and Nelson (1984)). Note that we do not restrict
our model to be quadratic in the wavevectors $\boldsymbol{q}$. Looking for
plane-wave solutions to the equations of motion gives the classical wave-
equation with general form:

$D(\boldsymbol{q})\boldsymbol{v}(\boldsymbol{q})=\omega^{2}(\boldsymbol{q})\boldsymbol{v}(\boldsymbol{q}),$
(3)

Where $\boldsymbol{q}=(q_{x},q_{y})$ is the wavevector of the plane-wave,
$D(\boldsymbol{q})$ is the dynamical matrix, whose topology we investigate,
$\omega^{2}(\boldsymbol{q})$ are the eigenfrequencies and
$\boldsymbol{v}(\boldsymbol{q})=(\boldsymbol{u},h)$. Note that a continuum
model can never capture optical branches in the phonon spectrum, as they
depend on the internal motion of atoms which we neglect. As we are only
interested in the topology of the acoustic phonons close to $\Gamma$ [e.g.
$\boldsymbol{q}=(0,0)$], the optical branches will have no impact on our
analysis. A more realistic model describing the phonons of graphene is
analyzed in Sec. V. For now, we think of $D(\boldsymbol{q})$ as $k\cdot p$
expansion, describing the acoustic bands around $\Gamma$, of the (many-band)
full phonon-band structure.

For stable structures, $D(\boldsymbol{q})$ is positive semidefinite, so that
$\omega$ is real. A more careful analysis of the constraints on
$D(\boldsymbol{q})$ is performed in Appendix A. For the Lagrangian in Eq. (1)
we find

$\small
D(\boldsymbol{q})\\!=\\!\begin{pmatrix}v_{l}^{2}q_{x}^{2}\\!+\\!v_{t}^{2}q_{y}^{2}&(v_{l}^{2}\\!-\\!v_{t}^{2})q_{x}q_{y}\\!&\\!0\\\
(v_{l}^{2}\\!-\\!v_{t}^{2})q_{x}q_{y}&v_{l}^{2}q_{y}^{2}\\!+\\!v_{t}^{2}q_{x}^{2}\\!&\\!0\\\
0\\!&0\\!&v_{h}^{2}(q_{x}^{4}\\!+\\!2q_{x}^{2}q_{y}^{2}\\!+\\!q_{y}^{4})\end{pmatrix}.$
(4)

Solving the eigenvalue problem in Eq. (3) gives explicitly

$\omega_{1}^{2}=v_{h}^{2}\boldsymbol{q}^{4},\quad\omega_{2}^{2}=v_{l}^{2}\boldsymbol{q}^{2},\quad\omega_{3}^{2}=v_{t}^{2}\boldsymbol{q}^{2}.$
(5)

The associated eigenvectors then read

$\boldsymbol{v}_{1}=\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix},\quad\boldsymbol{v}_{2}=\frac{1}{|\boldsymbol{q}|}\begin{pmatrix}q_{x}\\\
q_{y}\\\
0\end{pmatrix}\quad\boldsymbol{v}_{3}=\frac{1}{|\boldsymbol{q}|}\begin{pmatrix}-q_{y}\\\
q_{x}\\\ 0,\end{pmatrix}$ (6)

where $v_{l}=\sqrt{(2\mu+\lambda)/\rho_{0}}$, $v_{t}=\sqrt{\mu/\rho_{0}}$, and
$v_{h}=\sqrt{\kappa_{0}/\rho_{0}}$ are the longitudinal, transverse and out-of
plane velocities respectively. We therefore get a triple degeneracy at
$\boldsymbol{q}=(0,0)$ with $\omega=0$, and with two linear bands and one
quadratic band crossing as shown in Fig.1. The quadratic band corresponds to
the out-of plane flexural mode, and it is well-known Jiang  _et al._ (2015);
Rudenko  _et al._ (2019); Taheri  _et al._ (2021) that such bands are
generically present in 2D materials. This quadratic band distinguishes the 2D
case from the 3D case studied in Ref. Park  _et al._ (2021). We note that the
flexural band is completely decoupled from the in-plane modes. This is not
just a feature of our simplified model: the bands remain decoupled as long as
the harmonic approximation remains valid (e.g. we can ignore phonon-phonon
couplings). This is also the case with the models for graphene considered in
Sec. V, which have a similar decoupling. In graphene, this can also be
understood as arising from the fact that the flexural and in-plane bands have
opposite eigenvalues under the horizontal mirror operation Mañes (2020). We
will argue below that this decoupling, a feature of the 2D case, is intimately
tied to the nodal charge of the triple point.

We finally note that the bands $\boldsymbol{v}_{2}$ and $\boldsymbol{v}_{3}$
respectively correspond to a curl-free radial vector field and a divergence-
free angular vector field, as illustrated in Fig. 1c) and d). This simplifies
computation, but is not a generic feature of flexural phonons.

Figure 1: Summary of the dynamical matrix specified in Eq. (4). a) Band
structure in 3D with $v_{t}=v_{h}=1$ and $v_{l}=2$, b) Same as a) but along
the line $q_{x}=q_{y}$, where the lowest band (blue) is the flexural band, c)
vector field corresponding to the eigenvector
$\boldsymbol{v}_{2}=|\boldsymbol{q}|^{-1}(q_{x},q_{y},0)$, see Eq. (4), d)
vector field corresponding to the eigenvector
$\boldsymbol{v}_{3}=|\boldsymbol{q}|^{-1}(-q_{y},q_{x},0)$

.

###  II.2 Symmetry considerations in 2D

As mentioned in the introduction, we assume throughout that our models are
non-magnetic (have spinless time-reversal symmetry $\mathcal{T}$) and have a
generalized inversion symmetry $\mathcal{P}$, which satisfy
$(\mathcal{PT})^{2}=+1$. The physical constrains make these assumptions valid
close to $\Gamma$ quite generally, as discussed in Appendix A. We represent
the antiunitary symmetry as $\mathcal{T}=\mathcal{UK}$, where $\mathcal{U}$ is
the (unitary) orbital action and $\mathcal{K}$ is complex conjugation. Then
$\mathcal{P}$ and $\mathcal{T}$ act on $D(\boldsymbol{q})$ as:

$\mathcal{U}D^{*}(-\boldsymbol{q})\mathcal{U}^{-1}=D(\boldsymbol{q}),\quad\mathcal{P}D(-\boldsymbol{q})\mathcal{P}^{-1}=D(\boldsymbol{q}).$
(7)

This, together with the assumption that $(\mathcal{PT})^{2}=+1$, implies that
there exists a basis in which $\mathcal{PT}=\mathcal{K}$, so that
$D(\boldsymbol{q})$ can be chosen to be a real symmetric matrix Bouhon  _et
al._ (2020b). In the language of Ref. Bzdušek and Sigrist (2017), we are in
AZ+$\mathcal{I}$ class AI. For the model in Eq. (4),
$\mathcal{U}=\mathcal{P}=\mathbb{1}$.

As remarked in the introduction, there are no 3D irreducible representations
(IRREPs) in 2D layer groups in AZ class AI. Therefore, the triple point at
$\Gamma$ must consist of at least two IRREPs, which are glued together by the
NG theorem. This gluing of IRREPs is not protected by symmetry in 2D. This can
also be seen from a codimension argument Bzdušek and Sigrist (2017):
$\mathcal{PT}$ symmetry forces $D(\boldsymbol{q})$ to be an element of SO(3),
which is generated by the three rotation matrices $L_{i=x,y,z}$, The triple
band touching then requires tuning three independent parameters, but there are
only two momentum components available to tune. Therefore, this triple
crossing is cannot be stable in general, and only arise due to the NG theorem.
We confirm this in Sec. V by showing that when the NG theorem is modified by
adding a substrate, the triple degeneracy is lifted to a double degeneracy.
This agrees with our analysis in Sec. IV, where we show that this double
degeneracy has an associated non-trivial nodal charge. This illustrates that
the NG theorem can impose constraints on the band structure beyond any
symmetry formulation.

##  III Topological analysis from homotopy perspective

In this section, we investigate the topology associated with the nodal point
between the acoustic bands at $\boldsymbol{q}=\boldsymbol{0}$. Topological
charges of nodal points can generally be diagnosed by considering the homotopy
group of the classifying space Nakahara (2003).

To find the classifying space of our model, we note that the first Lamé
parameter in Eq. (1) satisfies $\mu>0$ Landau  _et al._ (1986). The second
Lamé parameter $\lambda$ can be negative, but is positive for most materials
Chicone (2017); Sadd (2021). We therefore generically expect $v_{l}>v_{t}$. As
we are working with an elastic continuum model, our model is only valid when
the wavelengths we are considering are much larger than the inter-atomic
spacing, which corresponds to small $\boldsymbol{q}$. In this limit, we expect
$\omega_{1}<\omega_{2}<\omega_{3}$ away from $\boldsymbol{q}=\boldsymbol{0}$,
so that we are considering three separate phonon branches (a $1\oplus 1\oplus
1$ split). This should be contrasted with the continuum model in 3D (see Park
_et al._ (2021) and Appendix B.1), where there are three linear bands, two of
which are degenerate (a $2\oplus 1$ split). In 2D, additional symmetries may
force the two linear bands to become degenerate along high-symmetry lines,
resulting in a $2\oplus 1$ split.

Because of our assumed $\mathcal{PT}$ symmetry, we can always choose
$D(\boldsymbol{q})$ to be a real symmetric matrix (see Sec. II.2 and Appendix
A), such that its eigenvectors
$(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\boldsymbol{v}_{3})$ are real and
their collection, i.e. the frame of eigenvectors, forms an element of O(3).
Under the reality condition, each eigenvector has a $\pm 1$ sign as a gauge
freedom. We can thus always locally choose a gauge where the frame has
positive determinant, i.e. it is an element of SO(3). Dividing out by the
group of gauge transformations that preserve the energy ordering of the bands,
as well as the handedness of the frame, we obtain the classifying space
$\mathrm{Fl}^{\mathbb{R}}_{1,1,1}=\mathrm{SO(3)/S[O(1)\times O(1)\times
O(1)]}$ for the $1\oplus 1\oplus 1$ split Bouhon  _et al._ (2020a). This is
the (unoriented) real complete Flag variety. It is also convenient to consider
the group of gauge transformations as the sign-exchange of each pair of
eigenvectors, i.e. the group of $\pi$-rotations along each of the three
eigenvectors, that is the point group
$D_{2}=\\{E,C_{2,\boldsymbol{v}_{1}},C_{2,\boldsymbol{v}_{2}},C_{2,\boldsymbol{v}_{3}}\\}$,
and the classifying space takes the compact form
$\mathrm{Fl}^{\mathbb{R}}_{1,1,1}=\mathrm{SO(3)/D_{2}}$ Wu  _et al._ (2019).
For the $2\oplus 1$ split, the classifying space reduces to the real
(unoriented) Grassmannian
$\mathrm{Gr}^{\mathbb{R}}_{2,3}=\mathrm{SO(3)/S[O(2)\times O(1)]}$, which is
isomorphic to the real projective plane, i.e.
$\mathrm{Gr}^{\mathbb{R}}_{2,3}\simeq\mathbb{R}\mathrm{P}^{2}$.

The topological charge of a nodal point in $D$ dimensions for a system with
classifying space $M$ is generically captured by the homotopy group
$\pi_{D-1}(M)$ Bzdušek and Sigrist (2017). These groups can be computed using
long exact sequences, as described in Hatcher (2001). The results given in
Ref. Bzdušek and Sigrist (2017); Wu  _et al._ (2019) are summarized in Tab. 1.
As we are only interested in the local topology of the node, we only need to
consider base loops and base spheres, such that the homotopy groups are
sufficient to classify the topological nodal phases. Indeed, global
topologies, i.e. over the whole Brillouin zone $\mathbb{T}^{D}$, requires the
consideration of homotopy equivalence classes $[\mathbb{T}^{D},M]$ which can
have more structure, such as nontrivial lower dimensional topologies over the
non-contractible cycles of the Brillouin zone torus, as e.g. the first
Stiefel-Whitney class Ahn  _et al._ (2018), computed along a full lattice
vector, and the action of the generators of $\pi_{1}[M]$ on the second
homotopy group Bouhon  _et al._ (2020a); Tiwari and Bzdušek (2020); Wojcik
_et al._ (2020). It follows that the question of orientability for
$d>1$-dimensional topologies is not relevant for us since the continuous maps
$\mathbb{S}^{d>1}\rightarrow M$ always induce an orientation, e.g. any mapping
$\mathbb{S}^{2}\rightarrow\mathbb{R}\mathrm{P}^{2}$ can be decomposed into a
winding component $\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$ and an orientable
double cover $\mathbb{S}^{2}\rightarrow\mathbb{R}\mathrm{P}^{2}$ Bouhon  _et
al._ (2020a).

We note that some entries in Tab. 1 capture fragile topology, in the sense
that adding additional trivial bands can change their value. The
$\mathbb{Z}_{2}$ charge (corresponding to the first Stiefel-Whitney class Ahn
_et al._ (2018, 2019a)) is stable under the addition of trivial bands. The
quaternion charge $\mathbb{Q}$ turns into the $N$th Salingaros group under
addition of further bands Wu  _et al._ (2019); Bouhon  _et al._ (2020b).
Finally, the $2\mathbb{Z}$ charge (corresponding to the Euler class Ahn  _et
al._ (2018, 2019b); Bouhon  _et al._ (2020b, a), see below) turns into a
$\mathbb{Z}_{2}$ charge, the second Stiefel-Whitney class, under the addition
of additional bands Ahn  _et al._ (2018, 2019a). As we are only concerned with
the acoustic bands, this low-band limit is justified.

We finally note that the 3D topology of a nodal point, characterized by the
topology over a sphere wrapping the node, was considered in Ref. Park  _et
al._ (2021) for the $2\oplus 1$ split, in which case it classified by the
Euler class. Note that, as discussed there, the presence of this split
requires that the condition $v_{l}>v_{t}$ be satisfied along the high-symmetry
lines emanating from $\Gamma$. Otherwise, the three bands cannot be split on
any sphere surrounding $\Gamma$, so that there is no nodal charge (since then
SO$(3)$ gauge transformations are allowed, thus trivializing the classifying
space $\mathrm{SO(3)/SO(3)}=1$ 111We note that the nontrivial element of
$\pi_{1}[\mathrm{Gr}^{\mathbb{R}}_{3,N\geq 4}]=\mathbb{Z}_{2}$ Bzdušek and
Sigrist (2017) would correspond to a nodal point with a $\pi$-Berry phase
connecting the three acoustic bands with higher bands, contrary to the
assumption that the acoustic bands are separated from all the other bands in
the vicinity of $\Gamma$.). In contrast, for 2D phonons, the topology is
always well defined. Sufficiently close to $\Gamma$, the flexural mode will
always be at lower frequency than the in-plane modes, owing to the quadratic
dispersion. Thus, violating the condition that $v_{l}>v_{t}$ along high-
symmetry lines can only change the split from $1\oplus 1\oplus 1$ to $2\oplus
1$ in 2D. As can be seen in Tab. 1, this results in a reduction of the nodal
charge from $\mathbb{Q}$ to $\mathbb{Z}_{2}$, but it does not a priori
completely remove the topology (see, however, Sec. IV.2 for a caveat to this).
Therefore, the nodal charge in 2D is actually more stable than its 3D
counterpart as it can be defined in all 2D systems with $\mathcal{PT}$
symmetry. In 1D (discussed briefly in Appendix B.2) there is no stable nodal
charge.

Name | $M$ |  $\pi_{0}$($M$) |  $\pi_{1}$($M$) |  $\pi_{2}$($M$)
---|---|---|---|---
Fl${}^{\mathbb{R}}_{1,1,1}$ | SO(3)/D2 | $\mathbb{0}$ | $\mathbb{Q}$ | $\mathbb{0}$
$\mathrm{Gr}^{\mathbb{R}}_{2,3}$ |  $\mathbb{R}$P2 | $\mathbb{0}$ | $\mathbb{Z}_{2}$ | 2$\mathbb{Z}$
Table 1: Possible charge of triple point for acoustic phonons of various
dimensions. $\mathbb{Q}$ denotes the quaternion group. The $\mathbb{Z}_{2}$
charge corresponds to the first Stiefel-Whitney class on a loop around the
nodal point Ahn  _et al._ (2019a) and the $2\mathbb{Z}$ charge corresponds to
the Euler class on a sphere surrounding the nodal point (discussed in Ref.
Park  _et al._ (2021)).

##  IV Topology of 2D acoustic phonons

We now consider the topology of the 2D case in further detail. In 2D, the only
possible homotopy classifications are $\pi_{p}(X)$ for $p\in\\{0,1,2\\}$
Bzdusek and Sigrist (2017). $\pi_{2}$ charges correspond to considering
monopoles encapsulated by a surface, e.g. the Brillouin zone (BZ) or patches
thereof. These are therefore irrelevant to the nodal charges as they are
classified by loops around nodes. Furthermore, as can be seen in Tab. 1, the
$\pi_{0}$ charge is zero in all symmetry settings. Thus, the only relevant
invariant is the $\pi_{1}$ charge, which corresponds to taking a circle around
the triple point at $\Gamma$. Depending on whether the bands split as $2\oplus
1$ or $1\oplus 1\oplus 1$ over this circle, the relevant groups are either
$\mathbb{Z}_{2}$ or the quaternion group $\mathbb{Q}$ (see Tab. 1). We
investigate both charges in this section. We assume throughout that the three
acoustic bands are separated in energy from all other bands on a circle around
the triple point at $\Gamma$, and on the entire disc enclosed by this circle.

###  IV.1 Quaternion charge of the complete Flag variety

When the bands split as $1\oplus 1\oplus 1$, the relevant $\pi_{1}$ charge is
the quaternion group $\mathbb{Q}$. We are therefore a priori dealing with non-
abelian nodal charges. Non-abelian charges in band structures is a novel but
quickly growing field Wu  _et al._ (2019); Bzdusek and Sigrist (2017); Bouhon
_et al._ (2020b); Ünal  _et al._ (2020); Bouhon  _et al._ (2020a); Jiang  _et
al._ (2021); Tiwari and Bzdušek (2020); Peng  _et al._ (2021); Chen  _et al._
(2021); Guo  _et al._ (2021). For the quaternion group, there are five
conjugacy classes of stable nodal charges: $\\{1,-1,\pm i,\pm j,\pm k\\}$,
which correspond to combinations of nodes in various gaps Tiwari and Bzdušek
(2020); Jiang  _et al._ (2021). Here $i,j,k$ satisfy
$i^{2}=j^{2}=k^{2}=ijk=-1$. In general, the charges $i,j,k$ are only defined
up to equivalence because their sign is gauge dependent Jiang  _et al._
(2021), as discussed further in Sec. IV.1.2.

Such charges are usually encountered in the context of multi-gap systems,
where they are computed using the Euler class Nakahara (2003), discussed in
Sec. IV.1.2. The Euler class can be used to assign a charge to any two-band
systems and the non-abelian topology then leads to non-trivial braiding
statistics for nodes in various band gaps. We note that this is not the case
in our system: the triple point is pinned by the NG theorem, and this non-
abelian charge is therefore associated only with the triple point. The Euler
classification of triple points was briefly discussed in Ref. Jiang  _et al._
(2021), where the resultant charge of the triple point is determined by
knowing which (two-band) nodal points come together to form the triple point.
This requires splitting the triple degeneracy into two-band nodal points,
which is not generically possible for the acoustic phonon case due to the NG
theorem.

There are two ways around this. One possibility is to consider the frame-
rotation charge discussed in Johansson and Sjöqvist (2004); Bouhon  _et al._
(2020b). This construction can distinguish between the quaternion charges
$+1$, $-1$ and $\\{\pm i,\pm j,\pm k\\}$, but cannot distinguish between $\pm
i,\pm j$ and $\pm k$. Physically, this corresponds to knowing whether there
are no protected nodes (+1), protected single nodes between any of the bands
$(\pm i,\pm j,\pm k)$ or a protected double node ($-1$) Tiwari and Bzdušek
(2020); Jiang  _et al._ (2021). However, this charge gives no information
about which bands are involved in the nodal structure. We discuss the frame
rotation charge in Sec. IV.1.1.

Alternatively, we can introduce terms in the Hamiltonian which artificially
break the triple point, compute the charge of the resulting nodes, and then
construct a continuous path back to the triple point. In this case, the charge
of the triple point can be determined from the combination of charges of the
two-band nodal points. This can distinguish between $\pm i$, $\pm j$ and $\pm
k$, but requires constructing such a splitting. This is introduced in Sec.
IV.1.2. In that section, we also show that the frame rotation charge suffices
for systems with $\mathcal{T}$ (and $\mathcal{PT}$) symmetry. Such a splitting
procedure may, however, have physical relevance for 2D systems on a substrate,
as we discuss in Sec. V. A final alternative to characterize this charge has
recently been introduced in Ref. Wu  _et al._ (2019), using a lifting of the
frame charge calculation from the orthogonal group to the spin group. This
lifting map is discussed in Appendix C.

Finally, we also relate to the more familiar notion of Berry phase in Sec.
IV.1.3, and show that it is insufficient to capture the topology.

####  IV.1.1 Frame rotation charge

The frame rotation charge can in general distinguish between the conjugacy
classes $\\{1,-1,\\{\pm i,\pm j,\pm k\\}\\}$, but cannot distinguish between
the (gap-dependent) charges $\\{\pm i,\pm j,\pm k\\}$. We briefly introduce
this method here, and refer to Johansson and Sjöqvist (2004); Bouhon  _et al._
(2020b) for further discussion.

The frame rotation charge measures the ability of nodes within a simply
connected surface to annihilate. It derives from
$\pi_{1}[\mathrm{SO}(N)]=\mathbb{Z}_{2}$ for $N\geq 2$. For the specific case
of $N=3$ (the case of general $N$ is discussed in Refs. Wu  _et al._ (2019);
Bouhon  _et al._ (2020b)), consider a matrix of three ordered orthonormal
vectors (usually called a frame)
$F=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\boldsymbol{v}_{3})$. In our case,
these vectors correspond to the eigenstates of the acoustic bands. Moving
along a closed trajectory $\Delta$ in the BZ, we induce a mapping
$R(\boldsymbol{q})=F(\boldsymbol{q})^{T}F(\boldsymbol{q}_{0})$, with
$\boldsymbol{q}_{0}$ a fixed reference point, and $\boldsymbol{q}$ traversing
$\Delta$. By deforming $\Delta$ to a loop, this becomes a map from $S^{1}$ to
the space of frames, paramterized by an angle $\theta\in[0,2\pi]$. This
mapping can be decomposed into the basis elements $\\{L_{i}\\}_{i=x,y,z}$ of
the Lie algebra $\mathfrak{so}(3)$, as
$R(\theta)=\exp(\sum_{i=x,y,z}\varphi_{i}(\theta)L_{i})$. The accumulated
frame rotation charge is then:

$\varphi(\theta)=\sqrt{\sum_{i=x,y,z}\varphi_{i}(\theta)^{2}}$ (8)

If we require the frames to be completely equivalent after traversing
$\Delta$, then the entire trajectory $R(\theta)$ lies in SO(3) and
$\varphi(2\pi)=2\pi n$ for $n\in\mathbb{Z}$. By using the connection to the
spin group, one can show Bouhon  _et al._ (2020b) that $\varphi$ is periodic
modulo $4\pi$, in analogy to the Dirac belt trick. This agrees with
$\pi_{1}[\mathrm{SO}(3)]=\mathbb{Z}_{2}$, and shows that there are two
possible charges $\varphi(2\pi)=\\{0,2\pi\\}$ mod $4\pi$. If, however, we
allow the final frame to differ from the initial frame by a sign change of two
eigenvectors, then $R(\Delta)\in\mathrm{SO}(3)/\mathbb{Z}_{2}$, and
$\varphi(2\pi)=\pi$ mod $4\pi$ becomes a possible solution
($\varphi(2\pi)=3\pi$ only differs from $\varphi(2\pi)=\pi$ by a gauge
transformation).

To discuss the physical interpretations of $\varphi$, we introduce some
standard terminology for three-band systems Wu  _et al._ (2019); Bouhon  _et
al._ (2020b). We refer to the gap between the lowest-energy band and the
middle band as the principal gap, and a node in this gap is therefore a
principal node. Similarly, the gap between the middle band and the highest-
energy band is referred to as the adjacent gap, and nodes in this gap are
adjacent nodes. Note that these concepts are ill-defined for the triple
degeneracy, but become well-defined once we imagine infinitesimally splitting
the triple degeneracy as discussed in Sec. IV.1.2.

If there are no stable nodes between the eigenstates the constitute
$F(\boldsymbol{q})$ on or inside the trajectory $\Delta$, then the frame is
smooth everywhere and $\varphi(2\pi)=0$ mod $4\pi$. This corresponds to the
trivial quaternion charge $+1$. If there is a stable double node in either the
principal or the adjacent gap, then the frame must perform a $2\pi$ rotation
around the node, so that $\varphi(2\pi)=2\pi$ mod $4\pi$ corresponds to
quaternion charge $-1$. Finally, if there is a simple node in the principal
gap or the adjacent gap or in both gaps the frame performs a $\pi$ rotation so
that $\varphi(2\pi)=\pi$ mod $4\pi$ corresponds to quaternion charges $i,j,k$.
Thus, the frame rotation charge captures the stability of nodes, but does not
capture in which gap the nodes are located. This is addressed further in Sec.
IV.1.2 and Appendix C.

Concretely, for the continuum model in Eq. (4), the mapping $R$, formed by the
eigenstates of the model, is given by:

$R(\boldsymbol{q})=F^{T}(\boldsymbol{q})F(\boldsymbol{0})=\frac{1}{|\boldsymbol{q}|}\begin{pmatrix}\boldsymbol{|q|}&0&0\\\
0&q_{x}&q_{y}\\\ 0&-q_{y}&q_{x}\end{pmatrix},$ (9)

where we have ordered the eigenstates by energy and we have chosen a smooth
gauge. Parameterizing $R$ by planar polar coordinates $(r,\theta)$ then gives

$R(\theta)=e^{-\theta L_{x}},$ (10)

Note that this corresponds to a rotation around a fixed axis. This is
something we expect to hold more generally, due to the decoupling of the
flexural mode from the in-plane modes, discussed in Sec. II.1. When traversing
the entire loop, Eq. (10) gives $\varphi(2\pi)=2\pi$ mod $4\pi$, which gives a
quaternion charge of $-1$, indicating that there is a stable double node at
$\Gamma$ in the continuum model.

####  IV.1.2 Quaternionic charge and Euler class

Whilst the frame rotation charge suffices to determine whether or not there is
a protected nodal charge associated with the bands, it does not distinguish
nodes in different gaps. This is significant, as it is well-known that the
flexural mode at $\Gamma$ can be gapped away from zero frequency under certain
conditions Aizawa  _et al._ (1990); Al Taleb and Farías (2016). This happens
for graphene grown on certain substrates, and the magnitude of this splitting
is sometimes used as a rough indicator of the interaction between the
substrate and the graphene layers Al Taleb and Farías (2016); Zhao  _et al._
(2018); Zhang  _et al._ (2021). The substrate modifies the out-of plane
symmetry breaking, so that the NG theorem can not be straightforwardly
applied, and therefore the triple degeneracy is not required. Note that if the
interaction with the substrate is sufficiently weak, and the acoustic bands
remain separated from all other bands, them the nodal charge of free-standing
graphene should still be applicable to the case of graphene on a substrate. We
show in Sec. V that this nodal charge is non-trivial, suggesting that the
nodal charge only protects the crossing of the in-plane bands. To investigate
this further, we now discuss how to distinguish the charge of nodes in
different gaps using the Euler class Nakahara (2003); Hatcher (2001); Ahn  _et
al._ (2019b); Bouhon  _et al._ (2020b).

The Euler class is defined for two-band subspaces of three-band real
Hamiltonians, in analogy to the more familiar Chern class for complex
Hamiltonians. The Chern class is an integer obtained by integrating the Berry
curvature (a differential two-form) over closed, even dimensional manifolds.
Similarly, the Euler class between states $|v_{1}(\mathbf{q})\rangle$ and
$|v_{2}(\mathbf{q})\rangle$ is an even integer obtained by integrating the
Euler form,

$\textrm{Eu}(\boldsymbol{q})=\mathinner{\langle{\boldsymbol{\nabla}v_{1}(\boldsymbol{q})}|}\times\mathinner{|{\boldsymbol{\nabla}v_{2}(\boldsymbol{q})}\rangle},$
(11)

over closed even dimensional manifolds. In fact, the Euler form can be
understood as the Berry curvature of the state
$|v_{1}(\boldsymbol{q})\rangle+i|v_{2}(\boldsymbol{q})\rangle$ Bouhon  _et
al._ (2020b). The Euler class is only defined over orientable manifolds, but
this is not a problem for continuum model as there are no non-contractible
loops in the plane.

Note that the only closed, even dimensional manifold available in 2D is the
whole BZ, which makes it difficult to compute these quantities in a continuum
model (where the BZ corresponds to all of $\mathbb{R}^{2}$). However, for the
Euler class, a patch formulation exists (e.g. it is possible to compute it on
a subset of $\mathbb{R}^{2}$) as discussed in Refs. Ahn  _et al._ (2019b);
Bouhon  _et al._ (2020b); Jiang  _et al._ (2021); Peng  _et al._ (2021); Chen
_et al._ (2021). The patch Euler class over a patch $\mathcal{D}$ is defined
as

$\chi(\mathcal{D})=\frac{1}{2\pi}\left[\int_{\mathcal{D}}\mathrm{Eu}-\oint_{\partial\mathcal{D}}a\right]\in\mathbb{Z}$
(12)

Where $\partial{\mathcal{D}}$ is the boundary of $\mathcal{D}$. Furthermore,
$\mathrm{Eu}$ is the Euler 2-form in Eq. (11) which can alternatively be
defined as $\mathrm{Eu}=d\mathrm{a}=d\mathrm{Pf}\mathcal{A}$, where
$\mathcal{A}_{ij}=\langle
v_{i}(\boldsymbol{q})|dv_{j}(\boldsymbol{q})\rangle=\boldsymbol{A}_{ij}\cdot
d\boldsymbol{q}=\sum_{\alpha=x,y}\langle
v_{i}(\boldsymbol{q})|\partial_{q_{\alpha}}v_{j}(\boldsymbol{q})\rangle
dq_{\alpha}$ in terms of the band indices $i,j\in\\{1,2\\}$. The second term
in Eq. (12) then amounts to the integral of the Euler connection 1-form
$\mathrm{a}=\mathrm{Pf}\boldsymbol{A}\cdot d\boldsymbol{q}$. We note that this
definition intimately profits from the reality conditions of the two-band
Berry connections ensuring that it take values in the orthogonal Lie algebra
SO$(2)$. The integer $\chi(\mathcal{D})$ equals minus twice the number of
stable nodes between the two bands inside $\mathcal{D}$ Ahn  _et al._ (2019b).
This should be contrasted with the Chern class, where no patch formulation is
readily obtainable without gauge fixing, showing that the Euler class is an
ideal tool for analyzing continuum models.

One characteristic feature of Euler class topology is that there can be
multiple nodes in the same gap that are unable to annihilate. To correctly
capture this property, a consistent gauge assignment must be made. This is
done by drawing Dirac strings between any pair of nodes, which correspond to
branch cuts across which the gauge must change. Detailed rules for assigning
such strings can be found in Refs. Jiang  _et al._ (2021); Peng  _et al._
(2021). Most importantly, whenever a principal node (see previous section)
crosses the Dirac string of an adjacent node, or vice versa, its chirality
must flip. This leads to non-trivial braiding statistics and non-abelian
charges. Knowing which gap hosts stable nodes, one can then assign quaternion
charge $i$ for single nodes in the principal gap, $j$ for nodes in the
adjacent gap, $k$ for one node in both gaps and $-1$ for a double node in
either gap Wu  _et al._ (2019). Note that the signs of $i,j,k$ flip when
crossing a Dirac string Ahn  _et al._ (2019b), which explains the assertion
made above that the $i,j,k$ are only defined modulo a sign. The charge $-1$
consists of a double node and is therefore unaffected by crossing a Dirac
string, giving rise to the aforementioned $5$ equivalence classes.

The patch Euler class is only well-defined for two-band subspaces, so the
patch must be chosen so as to only contain either principal or adjacent nodes.
This is clearly not possible for the triple point. This can be circumvented by
artificially adding a term to our dynamical matrix which splits the triple
degeneracy into principal and adjacent nodes. If this splitting can be
adiabatically mapped back to the original triple point, then the charge of the
principal/adjacent nodes should reflect the charge of the triple point.

To make this concrete, we consider the continuum model of Eq. (4). If we
wanted to capture the physics of graphene on a substrate discussed in Refs.
Aizawa  _et al._ (1990); Al Taleb and Farías (2016) and Sec. V, we should lift
the flexural band up in frequency. However, this leads to nodal lines rather
than nodal points. We perform this lifting for graphene in Sec. V. Here, we
instead add an onsite energy to one of the orbitals contributing to the
linearly dispersing bands, modifying $D(\boldsymbol{q})$ from Eq. (4) as:

$\tilde{D}(\boldsymbol{q})=D(\boldsymbol{q})+\mathrm{diag}\\{\epsilon,0,0\\}.$
(13)

Note that because $v_{l}\neq v_{t}$ in our model, this will break $C_{4}$
invariance, but maintain $C_{2}T$ invariance (as well as $C_{2}$ and $T$
separately), and therefore the reality condition. This splits the triple point
into two adjacent nodes on the $q_{y}$-axis and a single principal node at
$\Gamma$. This is illustrated in Fig. 2b).

Figure 2: a) Schematic of the Euler class computation, following the graphical
notation from Ref. Jiang  _et al._ (2021). Perturbing the dynamical matrix in
Eq. (4) by an onsite term of magnitude $\epsilon$ creates two nodes in the
adjacent gap (blue triangles) with opposite chirality (empty/filled) connected
by a Dirac string, and one principal node in the center (brown cirlce) with
trivial charge. This is confirmed by computing the patch Euler class over the
green annulus ($\chi=-1$ and over the gray circle ($\chi=0$), b) The band
structure corresponding to the situation sketched in a). The code used is
available at Ref. Bzdusek (2020)

Computing the Euler class over an annulus/disc avoiding the principal/adjacent
node, using the code in Ref. Bzdusek (2020), we find that the principal node
at the center has $\chi=0$, corresponding to quaternion charge $+1$. The
adjacent nodes on the $q_{y}$-axis have a combined Euler class of $\chi=-1$,
giving a corresponding quaternion charge of $-1$. Combining the nodes by
taking $\epsilon\rightarrow 0$ gives that the total quaternionic charge is
$-1$, in agreement with what was found using the frame rotation charge in the
previous section. However, knowing the gap structure, we now know that this
charge is associated only with the crossing between the linear bands 222Note
that double nodes generically have quadratic band touchings associated with
them Jiang  _et al._ (2021). However, the eigenvalues of the dynamical matrix
$D(\boldsymbol{q})$ correspond to $\omega^{2}$, so that a quadratic eigenmode
in $D(\boldsymbol{q})$ correspond to a linear dispersion as a function of
$\omega$.. Thus, the crossing between the quadratic and the linear bands is
not topologically protected, whereas the crossing between the linear bands is
protected. Note that because the nodal charge originates from
$\pi_{1}[\mathrm{SO}(3)]$, this charge is actually stable in the many-band
limit. We emphasize that this node cannot be obtained from an irreducible
representation (IRREP analysis), as it is present even in layer groups without
2D IRREPs.

This result can be understood straightforwardly by noting that adding any
perturbation of the form $\mathrm{diag}\\{0,\delta,0\\}$ to
$\tilde{D}(\boldsymbol{q})$ (with $\delta>0$, e.g. adding another onsite term)
will completely remove the principal node. This is therefore an accidental
node, in the sense that it is not symmetry protected. It is, however,
protected by the NG theorem in as it cannot be removed without modifying the
conditions of the theorem (e.g. by adding a substrate). This also implies that
there are strong constraints preventing the lifting of the in-plane acoustic
modes, e.g. if Goldstone’s theorem is broken adiabatically, we only expect the
flexural band to gap out (though the crossing point between the linear bands
may shift in frequency).

We now argue why we generically expect nodal charges of $\\{+1,-1\\}$ for
acoustic phonons in systems that have $\mathcal{P}$ and $\mathcal{T}$ symmetry
separately (rather than just their product). As discussed in Sec. II.2 and
elaborated in Appendix A, this is in fact a very general condition when
sufficiently close to $\Gamma$, as a consequence of the physical constraints
on the phonons.

Time-reversal symmetry $\mathcal{T}$ implies that if there is a band touching
at $\boldsymbol{q}$, then there is also be a band touching, between the same
bands, at $-\boldsymbol{q}$. Let us assume without loss of generality that the
node at $\boldsymbol{q}$ has charge $+i$. Then the node at $-\boldsymbol{q}$
has charge $\pm i$, (the sign depends on the location of the Dirac strings
from the adjacent nodes) . Now imagine splitting the triple point into two
pairs of nodes in each gap (as required by the presence of $\mathcal{T}$). Let
us assign charge $\pm i$ to nodes in the first gap and $\pm j$ to nodes in the
second gap. The total node configuration at the triple point will then have
charge $Q=(\pm i)(\pm i)(\pm j)(\pm j)$, where the order of the factors
depends on the details of how the nodes are adiabatically brought together.
Regardless of the order, however, the only possible result is $Q=\pm 1$. Thus,
generically (that is, unless there is a symmetry beyond the NG theorem pinning
the nodes at $\Gamma$), we expect the quaternionic charge to reduce to $\pm
1$. Thus, the physical constraints can give topology beyond what is expected
from symmetry analysis, but they also constrains the nodal charge beyond the
symmetry analysis.

####  IV.1.3 Relating to Berry phase

We now relate the above findings to the more conventional Berry phase
formulation of nodal charges, and show that it is insufficient to capture this
topology.

For a single band in a system with generalized $\mathcal{PT}$ symmetry, the
only gauge freedom is a choice of sign. If the sign of the eigenvector
necessarily flips as it is transported around the loop, there must be a
discontinuity in the gauge somewhere along the loop, due to the discreteness
of the gauge group (this corresponds to the Dirac string discussed above).
This indicates that the band under consideration forms an odd number of
topologically protected nodes within the loop. As discussed in Ahn  _et al._
(2018, 2019a), such a discontinuity can be analyzed by using a smooth complex
gauge, where a Berry phase of $\pi$ indicates a sign reversal. Thus, along the
loop, each band can (in a smooth complex gauge) have a Berry phase of $0$ or
$\pi$.

As we assume the acoustic bands to be separated from all other bands on the
loop and the disc it encloses, the sum of the Berry phases of the three
acoustic bands must be $0$ mod $2\pi$ (because a single node induces a Berry
phase of $\pi$ in both bands forming the node). We write the Berry phases of
the bands as $\vec{\varphi}=(\varphi_{1},\varphi_{2},\varphi_{3})$, where we
have ordered the bands by their associated frequency on the loop. Thus, e.g.
the phases $(\pi,\pi,0)$ indicate a principal node, $(0,\pi,\pi)$ indicate an
adjacent node and $(\pi,0,\pi)$ indicate one principal and one adjacent node.
These respectively correspond to quaternion charge $i,j,k$. Note that the
Berry phase $(0,0,0)$ can correspond to quaternion charge $+1$ or $-1$, as the
Berry phase is oblivious to the presence of double nodes.

In the specific case of flexural phonons, we expect the lowest energy flexural
band to be completely decoupled from the other two bands, so that it
necessarily has a trivial Berry phase. Therefore, the only possible
assignments of Berry phase are $\vec{\varphi}=\\{(0,0,0),(0,\pi,\pi)\\}$. From
the above discussion, we generically expect a quaternion charge of $\pm 1$ in
systems with TRS, leaving only $\vec{\varphi}=(0,0,0)$. Thus, the nodal charge
of acoustic phonons in 2D is completely invisible to the Berry phase.

###  IV.2 $\mathbb{Z}_{2}$ charge of the real Grassmannian in 2D

The above discussion applies when all three bands can be split on a loop
around $\Gamma$. If there is a symmetry which forces the two linear bands to
be degenerate, either along high-symmetry lines or everywhere, then any loop
around the triple point will necessarily contain a node between the linear
bands. Thus, the classifying space is the real Grassmannian and the associated
$\pi_{1}$ charge in Tab. 1 is $\mathbb{Z}_{2}$, which corresponds to the first
Stiefel-Whitney class as discussed in Ref. Ahn  _et al._ (2019a). This
invariant measures the orientability of the real wavefunction as one traverses
a loop. Specifically, this corresponds to whether or not there is
(necessarily) a sign reversal of the subframe spanned by the bands under
consideration. This number can be defined for either the flexural mode or the
two linearly dispersing modes. For the flexural mode, it corresponds to the
Berry phase computed in a smooth complex gauge as also discussed in Sec.
IV.1.3. Note that, because there is no coupling between the flexural and the
linearly dispersing bands, the flexural mode is trivial, and there therefore
exists an obvious gauge where the orientation is constant. Thus, the first
Stiefel-Whitney invariant is trivial, and there is no protected $\pi_{1}$
charge for this symmetry setting. Note that this argument holds generally for
dynamical matrices, not just in the continuum model, due to the decoupling of
the flexural band non-interacting limit discussed in Sec.II.1.

This argument relies on a global cancellation condition - e.g. that the Berry
phase of all three bands must be zero, since these bands are disconnected from
the other bands at higher energy (indeed, a resultant $\pi$-Berry phase
indicates an unavoidable node with the other bands). This is to be contrasted
with the quaternion charges of the frame, since the nontrivial frame charges
indicate stable nodes among the three bands of the frame and not with the
other bands. We furthermore note that a nontrivial quaternion charge of $-1$
(Euler class of $\pm 1$) around a region of the Brillouin zone is not required
to be cancelled by compensating nodes in any other region of the Brillouin
zone, contrary to the Berry phase and the quaternion charges $i,j,k$. This
directly implies that only an even number of nodes are allowed within each gap
when considering the whole Brillouin zone.

##  V Application: Graphene

In this section, we apply the above ideas to the paradigmatic 2D material
graphene. We show that the nodal charge described in the previous section is
non-trivial in this system, and that this charge predicts how phonons in
graphene will react to the presence of a substrate.

Previous work on phonon topology in free-standing graphene Kariyado and
Hatsugai (2015); Li  _et al._ (2020) have identified various topological nodal
points and lines in the spectrum away from $\Gamma$. Using methods from
topological quantum chemistry (TQC) Bradlyn  _et al._ (2017), Ref. Mañes
(2020) studied the symmetry decomposition of the in-plane phonon modes, and
found that in-plane phonons in graphene are globally trivial from the
perspective of TQC, though they are close to a fragile phase.

The previous topological analyses do not address the acoustic triple point at
$\Gamma$. However, we now show that this triple point crossing with the
flexural band actually possesses a non-trivial nodal charge (the frame-
rotation charge), which to the best of our knowledge has not been reported
before.

There exist a variety of models for graphene, including valence force-field
models (VFFMs) Aizawa  _et al._ (1990); Jiang  _et al._ (2015), spring models
Kariyado and Hatsugai (2015) and symmetry-based tight binding models Falkovsky
(2007); Michel and Verberck (2008). We implement a VFFM for graphene as
described in Ref. Aizawa  _et al._ (1990). This model explicitly considers six
terms: nearest and next-nearest neighbor bond stretching, in-plane and out-of
plane bending, bond twisting and interactions with the substrate. The energy
associated with each of these terms is written in terms of the displacement of
the various atoms in the unit cell, giving a total energy $V$. In the harmonic
approximation, this is differentiated twice with respect to the possible
displacements of the atoms in the unit cell. As there are two atoms in the
unit cell, which can displaced in three independent directions, this gives a
total of six phonon branches. The strength of the various terms in the energy
are then treated as fitting parameters to experimental dispersion, as shown in
Ref. Aizawa  _et al._ (1990). The term describing interaction with the
substrate is zero for free-standing graphene, but non-zero when coupling to a
substrate. When this term is non-zero, the acoustic flexural bands gaps out
from the other acoustic bandsAizawa  _et al._ (1990); Al Taleb and Farías
(2016); Zhao  _et al._ (2018); Zhang  _et al._ (2021).

We consider the case of free-standing graphene as well as graphene on the
substrate TaC(111). For both cases, we implement the model described above and
solve it on a loop encircling $\Gamma$, ensuring that none of the three lowest
bands touch on the loop and that we are sufficiently close to $\Gamma$ to
avoid any interference from the three upper bands. Before solving, we rotate
the dynamical matrix to a real basis. We choose the gauge of the initial point
so that the matrix $F(\boldsymbol{q})$ in Sec. IV.1.1 has determinant $+1$. We
can then choose a smooth gauge, by choosing the sign of each eigenvector on
the loop so that it maximizes the overlap with the previous eigenvector.
Decomposing the matrix $R(\boldsymbol{q})$ in Sec. IV.1.1 into rotation
generators, we can then plot the accumulated angle, as shown in Fig. 3, where
we also plot the band structure. The model in Ref. Aizawa  _et al._ (1990) is
fitted only on the line $\overline{\Gamma\mathrm{M}}$, but as we are only
interested in a circle around $\Gamma$, this suffices for our purposes. Note
that there appears to be an additional triple point in the optical phonons at
$K$, but this is an artifact of using a model which is fit only on the line
$\overline{\Gamma\mathrm{M}}$. In the full first-principle phonon spectrum
Mounet and Marzari (2005), this triple point is absent.

Figure 3: Phonon band structure and frame rotation charge on a circle around
$\Gamma$ for graphene, based on a valence force-field model from Aizawa  _et
al._ (1990). a) Phonon bands along high-symmetry lines for free-standing
graphene. b) Frame rotation charge on a loop around $\Gamma$ for free-standing
graphene c)and d): Same as a) and b) for graphene on a TaC(111) substrate

Fig. 3 shows that the nodal charge for free-standing graphene and graphene on
TaC(111) is $-1$. We also see that this charge is associated with the
degeneracy between the linear bands, as the nodal charge does not change when
gapping the flexural band. We corroborate these result by repeating the above
calculation for free-standing graphene using the symmetry-based model found in
Ref. Falkovsky (2007). This leads to the same charge.

##  VI Conclusions

We have discussed how physical constraints for phonons interplay with symmetry
analysis. We summarized the possible nodal charges of acoustic phonons with a
reality condition in up to three dimensions in Tab. 1, and discussed in detail
how to compute and analyze these charges in 2D.

We have found that acoustic phonons in 2D have an effective inversion symmetry
close to $\Gamma$, imposed by physical constraints (see Appendix A). This
leads to acoustic phonons generically having a quaternionic charge, which
however is further modified by the physical constraints to be $\\{+1,-1\\}$.
Additionally, the physics dictates that one of the acoustic bands (the
flexural band) is completely decoupled from the other bands in the non-
interacting limit, which allowed us to identify the nodal charge as belonging
only to two of the bands.

Applying the above machinery to graphene, we showed that acoustic phonons in
graphene have a non-trivial nodal charge which has not been previously
addressed. Knowing that this charge is associated with only two of the bands
explains, from a purely topological perspective, the well-known fact that the
flexural band of graphene on a substrate can gap from the other acoustic
bands.

These points illustrate that symmetry constraints must in certain cases be
augmented by physical constraints for band structure analysis. We anticipate
that similar effects could arise in other physical contexts. One example may
be photonic lattices. In this context, optical responses necessarily feature a
Bosonic spectrum with an inherent ”particle-hole” symmetry, as well as
persistent zero modes that are rather similar to an acoustic mode. This can
also lead to triple nodes at zero frequency that are not rooted in symmetry
(in fact irreducible representations can not formally be assigned in this
case).

Finally, from a theoretical perspective it could be interesting to relate
above type of analyses to topological charges of non-linear sigma models. Such
sigma models have recently also been evaluated from a Flag manifold
perspective, as in this work (see e.g. Ref Kobayashi  _et al._ (2021)). It
would therefore be interesting to investigate whether these mathematical
results find solid ground in the context we have considered.

###### Acknowledgements.

We thank Bo Peng for useful discussions. G.F.L acknowledges funding from the
Aker Scholarship. B.M. acknowledges support from the Gianna Angelopoulos
Programme for Science, Technology, and Innovation and from the Winton
Programme for the Physics of Sustainability. R.-J. S. acknowledges funding
from the Marie Skłodowska-Curie programme under EC Grant No. 842901, the
Winton Programme for the Physics of Sustainability, and Trinity College at the
University of Cambridge.

##  Appendix A Further constraints on the dynamical matrix

In this section, we analyze the constraints on the dynamical matrix around
$\Gamma$ ($\boldsymbol{q}=\boldsymbol{0}$) which arise from constraints that
are not intrinsically captured by a pure space group (SG) analysis. These
constraints set the phonon problem apart from the corresponding electronic
problem. We mostly discuss the case with time-reversal symmetry, but briefly
comment on the magnetic case at the end.

###  A.1 Constraints with time-reversal symmetry

Let us begin by briefly reviewing the constraints that the Bloch Hamiltonian
$H(\boldsymbol{k})$ of a non-magnetic electronic system on a lattice should
satisfy. The only required symmetry operations in this setting are lattice
translations and time-reversal. Working perturbatively close to $\Gamma$, we
can work with an effective continuum (local) model
$\tilde{H}(\boldsymbol{k})$. The only constraints on permissible local
Hamiltonians are then unitarity, i.e.
$H(\boldsymbol{k})=H^{\dagger}(\boldsymbol{k})$ and time-reversal symmetry
TRS, i.e. $UH^{*}(-\boldsymbol{k})U^{\dagger}=H(\boldsymbol{k})$ for some
unitary operator $U$. Depending on whether spin-orbit coupling (SOC) can be
discarded or not, the TRS operator may square to $+1$ (-1), corresponding
respectively to AZ class AI or AII. Additional constraints on the Bloch
Hamiltonian may arise from spatial symmetries. Such constraints have been
extensively analyzed in the literature Bradley and Cracknell (1972); Fu
(2011); Slager  _et al._ (2012); Kruthoff  _et al._ (2017); Po  _et al._
(2017); Bradlyn  _et al._ (2017) and form the symmetry classification of Bloch
Hamiltonian, based on an analysis of space groups (SGs).

We now turn to describing phononic systems, and show that the same constraints
emerge, but that they are supplemented by additional conditions due to the
physical constrainst discussed in Sec. I. We consider the dynamical matrix in
the harmonic approximation, which in any dimensions is given by Maradudin and
Vosko (1968); Brüesch (1982):

$D_{\alpha\beta}(ss^{\prime}|\boldsymbol{q})=\frac{1}{\sqrt{m_{s}m_{s}^{\prime}}}\sum_{\vec{l}}\Phi_{\alpha\beta}(\vec{0}s;\vec{l}s^{\prime})e^{i\boldsymbol{q}\cdot\boldsymbol{x}(\vec{l})}$
(14)

Where $\alpha,\beta$ label the Cartesian coordinates, $s,s^{\prime}$ label the
atoms in the unit cell with masses $m_{s},m_{s^{\prime}}$, $\vec{l}$
enumerates the unit cells with coordinate $\boldsymbol{x}(\vec{l})$ and
$\Phi_{\alpha\beta}(\vec{0}s;\vec{l}s^{\prime})$ is the force constant matrix
in the harmonic approximation:

$\Phi_{\alpha\beta}(\vec{l}s;\vec{l}^{\prime}s^{\prime})=\frac{\partial^{2}V}{\partial
u_{\alpha}(\vec{l}s)\partial
u_{\beta}(\vec{l^{\prime}}s^{\prime})}\big{|}_{u=0}$ (15)

Here $V$ is the total potential energy, $u_{\alpha}(\vec{l}s)$ is the
displacement along $\alpha$ of atom $s$ in unit cell $\vec{l}$ and the matrix
is evaluated at the equilibrium position of the atoms. Because we expect that
$\Phi$ is real, we immediately find:

$D_{\alpha\beta}(ss^{\prime}|\boldsymbol{q})=D^{*}_{\alpha\beta}(ss^{\prime}|-\boldsymbol{q})$
(16)

Thus, we automatically satisfy spinless TRS in this formalism. (We provide a
brief overview of how to break TRS in phononic systems in Appendix A.2. A more
detailed discussion can be found in Ref. Liu  _et al._ (2020).) Furthermore,
by commuting the partial derivatives, we find:

$\Phi_{\alpha\beta}(\vec{l}s,\vec{l}^{\prime}s^{\prime})=\Phi_{\beta\alpha}(\vec{l^{\prime}}s^{\prime};\vec{l}s),$
(17)

which implies that
$D_{\alpha\beta}(ss^{\prime}|\boldsymbol{q})=D_{\beta\alpha}(s^{\prime}s|\boldsymbol{q})$.
As shown in Ref.Brüesch (1982), it follows that that $D$ is Hermitian. We note
that it is not generally true that
$D(\boldsymbol{q})=UD(\boldsymbol{-q})U^{\dagger}$ for some unitary $U$ (this
is condition is what we in the main text refer to as a generalized inversion
symmetry). Therefore, $D(\boldsymbol{q})$ is not in general unitarily
equivalent to a real matrix.

So far, all results are analogous to the non-magnetic electronic case, and
just like the electronic case, additional constraints can now arise from
crystalline symmetries. However, even without additional symmetries, there are
further constraints (for stable structures) on the form of $D(\boldsymbol{q})$
which are not present for the Bloch Hamiltonian $H(\boldsymbol{k})$. As
discussed in section I, $D(\boldsymbol{q})$ must be positive semidefinite to
avoid imaginary frequencies, which correspond to an unstable structure.
Furthermore, there should be an appropriate number of zero-energy acoustic
bands at $\boldsymbol{q}=\boldsymbol{0}$, as dictated by the NG theorem. We
here assume that the NG theorem is not modified by any substrate. It turns out
that these constraints are sufficient to guarantee that the nodal charge of
the acoustic bands at $\Gamma$ is always captured by real topology.

To show this, let us focus on some region around $\Gamma$ in the BZ.
Sufficiently close to $\Gamma$, we can construct an effective dynamical matrix
$\tilde{D}(\boldsymbol{q})$ containing only the acoustic bands, e.g. if there
are $N$ acoustic bands then $\tilde{D}(\boldsymbol{q})$ is an $N\times N$
matrix. This is guaranteed from the fact that the acoustic modes all go to
zero 333Note that this construction is not guaranteed to work for the
substrate case considered in Ref. V which breaks basal mirror symmetry.
Nevertheless, if the lifted flexural band does not cross any acoustic bands
close to $\Gamma$ then such an effective $\tilde{D}(\boldsymbol{q})$ still
exists for the remaining acoustic bands at $\Gamma$.. We require that all
eigenvalues of $\tilde{D}(\boldsymbol{0})$ are zero, so that
$\tilde{D}(\boldsymbol{0})=\mathbb{0}$. As this is an effective model, we do
not require it to be positive semidefinite everywhere. Instead, we only
require that it should be positive semidefinite on a ball $B_{\epsilon}$ of
radius $\epsilon$ in $\boldsymbol{q}$-space surrounding $\Gamma$, where we
also assumes that the acoustic bands stay detached from all optical bands on
$B_{\epsilon}$. We assume throughout that $\epsilon>0$. We can now expand
$\tilde{D}(\boldsymbol{q})$ in powers of $\boldsymbol{q}$ in $B_{\epsilon}$,
where we note that the condition $\tilde{D}(\boldsymbol{q})=\mathbb{0}$
precludes a constant term. We denote the (fixed) basis matrices for $N\times
N$ hermitian matrices by $\\{\Gamma_{i}\\}_{i=1}^{N^{2}}$ (these can be chosen
to be the identity and the Pauli matrices for $N=2$ and the identity and the
Gell-Mann matrics for $N=3$). The most general form of the dynamical matrix is
then:

$\tilde{D}(\boldsymbol{q})=\alpha_{1}^{jk}q_{j}\Gamma_{k}+\alpha_{2}^{lmn}q_{l}q_{m}\Gamma_{n}+\mathcal{O}(\boldsymbol{q}^{3}),$
(18)

where the summation convention for repeated indices has been assumed. Positive
semidefiniteness requires that:

$z\tilde{D}(\boldsymbol{q})z^{*}\geq 0\quad\forall z\in\mathbb{C}^{N}$ (19)

Let us now assume that the lowest order term that appears in the expansion is
of order $k$, and assume that $\tilde{D}(\boldsymbol{q})$ is positive definite
for $\boldsymbol{q}\in B_{\epsilon}$. Then, sufficiently close to $\Gamma$:

$\alpha_{k}^{i_{1}\dots i_{k}n}q_{i_{1}}\dots q_{i_{k}}z\Gamma_{n}z^{*}\geq
0\quad\forall z\in\mathbb{C}^{N}$ (20)

Fixing a $z\in\mathbb{C}^{N}\backslash\\{0\\}$, the same equation must hold at
$-\boldsymbol{q}$, which by assumption is also in $B_{\epsilon}$. This clearly
requires that $k$ be even, which gives the $k$th order term an effective
$\mathcal{PT}$ symmetry. Sufficiently cloes to $\Gamma$, only the term of
order $k$ will matter. Therefore, there will always be an effective
$\mathcal{PT}$ symmetry sufficiently close to $\Gamma$.

We note finally that the positive semidefinite condition does not further
constrain the number of permissible matrices $\\{\Gamma_{i}\\}$ that can
appear in $\tilde{D}(\boldsymbol{q})$ as it is always possible to choose a
basis for hermitian matrices consisting exclusively of positive semidefinite
matrices. TRS will in general constrain the number of basis matrices, but this
feature is shared between the phononic and the electronic case.

###  A.2 Breaking time-reversal symmetry in phononic systems

Phonons, as opposed to electrons, are electrically neutral. We therefore, a
priori, do not expect them to couple strongly to an external magnetic field,
and therefore breaking of TRS is a much more exotic effect in phonons than it
is in electrons. There are, however, various proposal to break TRS. Some ideas
include Raman spin-phonon couplings Ioselevich and Capellmann (1995); Zhang
_et al._ (2010), pseudo-magnetic fields induced by the Coriolis force Wang
_et al._ (2015a, b) and optomechanical interactions Peano  _et al._ (2015).
This clearly goes beyond the standard formulation of the dynamical matrix
discussed in the previous section, as this automatically incorporates TRS (see
Eq. (16)).

The way around this is to introduce extra terms in the Lagrangian. A summary
of these effects can be found in Liu  _et al._ (2020). However, for non-
interacting phononic band structures in the 80 layer groups in AZ class AI,
such effects do not occur. We therefore do not discuss the breaking of TRS
further in this work.

##  Appendix B Acoustic phonons in 3D and 1D

###  B.1 Acoustic phonons in 3D

The continuum model for acoustic phonons in 3D can be derived in a similar
fashion as the 2D model Landau  _et al._ (1986). There are no flexural modes,
and in the continuum model two of the linearly dispersing bands are
degenerate. Concretely, the dynamical matrix is (where now
$\boldsymbol{q}=[q_{x},q_{y},q_{z}]$):

$D(\boldsymbol{q})=\begin{pmatrix}v_{3,T}^{2}\boldsymbol{q}^{2}&(v_{3,L}^{2}-v_{3,T}^{2})q_{x}q_{y}&(v_{3,L}^{2}-v_{3,T}^{2})q_{x}q_{z}\\\
(v_{3,L}^{2}-v_{3,T}^{2})q_{x}q_{y}&v_{3,T}^{2}\boldsymbol{q}^{2}&(v_{3,L}^{2}-v_{3,T}^{2})q_{y}q_{z}\\\
(v_{3,L}^{2}-v_{3,T}^{2})q_{x}q_{z}&(v_{3,L}^{2}-v_{3,T}^{2})q_{y}q_{z}&v_{3,T}^{2}\boldsymbol{q}^{2}\end{pmatrix}$
(21)

where $v_{3,T}$ and $v_{3,L}$ are the transverse and longitudinal velocities
in 3D respectively. In terms of elastic parameters, these are given by
$v_{l}=\sqrt{(\lambda+2\mu)/\rho_{0}}$ and $v_{t}=\sqrt{\mu/\rho_{0}}$. The
explicit eigenfrequencies of this model are given by:

$\displaystyle\omega_{1}^{2}=v_{3,L}^{2}\boldsymbol{q}^{2}$ (22)
$\displaystyle\omega_{2}^{2}=v_{3,T}^{2}\boldsymbol{q}^{2}$ (23)
$\displaystyle\omega_{3}^{2}=v_{3,T}^{2}\boldsymbol{q}^{2}$ (24)

The topology of this model was considered in Ref. Park  _et al._ (2021). In
agreement with Tab.1, they find that it is characterized by an Euler charge
over a closed surface, as long as there is a gap between $\omega_{2}$ and
$\omega_{3}$ away from $\Gamma$. Adding symmetry constraints can force the
three bands to cross along high-symmetry lines emanating from $\Gamma$, which
prevents the definition of a topological charge. As discussed in Ref. Park
_et al._ (2021), this happens when $v_{L}$ and $v_{T}$ become
$\boldsymbol{q}$-dependent and change relative magnitude along high-symmetry
lines. By building more complicated models, it may also be possible to lift
the two-band degeneracy away from $\Gamma$, allowing a multi-gap partitioning
not discussed in Ref. Park  _et al._ (2021). However, as can be seen in Tab.
1, such a multi-gap system would have trivial charge.

###  B.2 Model for 1D acoustic phonons

For a 1D material, the three bands completely decouple Landau  _et al._
(1986). We choose a rod geometry, where the material is extended in the
$z$-direction, with a very small radius in the xy-plane. There is one subtlety
in the 1D case however: the displacement field $\boldsymbol{u}(z)$ can be
large even if the strain tensor
$\overset{\text{\scriptsize$\leftrightarrow$}}{u}_{ij}$ is small. This is the
case for torsional modes, which correspond to a twisting of the 1D material,
leading to a linear dispersion relation Landau  _et al._ (1986). This could
potentially correspond to a fourth, torsional, mode in rod geometries. This is
not explored here, where we focus only on the vibrations along the rod axis
and perpendicular to it. Following Ref. Landau  _et al._ (1986), we then get
(in the classical regime) the trivial model:

$D(q_{z})=\begin{pmatrix}v_{l}^{2}q_{z}^{2}&0&0\\\ 0&v_{x}^{2}q_{z}^{4}&0\\\
0&0&v_{y}^{2}q_{z}^{4}\end{pmatrix}$ (25)

The velocities $v_{x}$ and $v_{y}$ depend on the moment of mass in the $x$ or
$y$ direction. If the material has radial symmetry around the $z$-axis (the
extended axis), then $v_{x}=v_{y}$. Thus, once again, we can get either a full
split or a partial split, but we will never get a case where all three bands
are degenerate. However, as can be seen from Tab. 1, the homotopy charge is
trivial independent of the split.

##  Appendix C Non-abelian Wilson loops and the lifting map

###  C.1 The lifting map

For completeness, we also include a method for distinguishing all five
conjugacy classes of the quaternion group $\mathbb{Q}$, $\\{1,-1,\pm i,\pm
j,\pm k\\}$ without having to split the nodes as was done in Sec. IV.1.2 .
This method was introduced in Wu  _et al._ (2019). The idea is to lift the
SO(3) valued Wilson loop to an SU(2) valued version, being isomorphic to the
quaternions with unit norm. We start with the $\mathfrak{so}$(3) valued Berry
connection which in component reads:

$[A(\boldsymbol{q})_{a}]^{i}_{j}=\langle
u_{\boldsymbol{q}}^{i}|\partial_{\boldsymbol{q}_{a}}|u_{\boldsymbol{q}}^{j}\rangle$
(26)

with $a\in\\{x,y\\}$ and $i,j\in\\{1,2,3\\}$. Being $\mathfrak{so}$(3) valued,
$A(\boldsymbol{q})$ can be decomposed into the basis matrices of the Lie
algebra $\\{L_{i}\\}_{i=x,y,z}$. These can then be lifted to the Lie algebra
of the double cover $\mathfrak{su}$(2) by replacing $\\{L_{i}\\}_{i=x,y,z}$
with the corresponding Dirac matrices, which in this case equals
$-\frac{i}{2}\sigma_{x,y,z}$. Then, computing the standard Wilson loop:

$n_{\Delta}=\overline{\exp}\left(\oint_{\Delta}\boldsymbol{A}(\boldsymbol{q})\cdot\mathrm{d}\boldsymbol{q}\right)$
(27)

along a contour $\Delta$, gives $n_{\Delta}\in$ SU(2), which is isomorphic to
the quaternionic group $\mathbb{Q}$ with unit norm. We remark that care must
be taken when computing the exponential, as the different matrices in the
exponent do not generically commute.

For the simple model in Eq. (4), this quantity is easily computed and we show
in the next section that we get $n_{\Delta}=-\mathbb{1}$, in agreement with
what was found by the other computations in Sec. IV. For more complicated
systems, an approximation of this expression, using the Baker-Campbell-
Hausdorff formula, is given in Wu  _et al._ (2019). We numerically compute
this quantity for the graphene system considered in Sec. V, and find that it
always agrees with our observed frame rotation charge when only considering
the lower three bands. However, considering all six bands [e.g. lifting SO(6)
to Spin(6)$\simeq$SU(4)] gives a trivial charge, which shows that the three
optical bands also carry a non-trivial frame rotation charge. This originates
from the degeneracy of the optical bands at $\Gamma$, visible in Fig. 3.

###  C.2 Computing $n_{\Delta}$ for the continuum model

In this section we compute the non-abelian Wilson loop charge $n_{\Delta}$
explicitly for the simple model in Eq. (4). Once again ordering by energy we
get:

$\displaystyle A_{x}=\langle
u_{i}|\partial_{q_{x}}|u_{j}\rangle=\frac{1}{|\boldsymbol{q}|^{2}}\begin{pmatrix}0&0&0\\\
0&0&-q_{y}\\\ 0&q_{y}&0\end{pmatrix}=\frac{q_{y}}{|\boldsymbol{q}|^{2}}L_{x}$
$\displaystyle A_{y}=\langle
u_{i}|\partial_{q_{y}}|u_{j}\rangle=\frac{1}{|\boldsymbol{q}|^{2}}\begin{pmatrix}0&0&0\\\
0&0&q_{x}\\\ 0&-q_{x}&0\end{pmatrix}=-\frac{q_{x}}{|\boldsymbol{q}|^{2}}L_{x}$

We note that this agrees with our observation from Sec. IV.1.1 that, because
one band is decoupled, the eigenvectors are rotated around a fixed axis. We
perform the lift by replacing $L_{x}\rightarrow-\frac{i}{2}\sigma_{x}$. Then,
letting $\boldsymbol{q}$ be along a loop away from the origin gives:

$\boldsymbol{A}(\boldsymbol{q})\cdot
d\boldsymbol{q}=\frac{i}{2}\sigma_{x}d\theta$ (28)

This is obviously independent of $\boldsymbol{q}$, so every matrix in the
exponential commutes. Letting $\Delta$ be a circle in the BZ, we then find:

$n_{\Delta}=\overline{\exp}\left(\oint_{\Delta}\boldsymbol{A}(\boldsymbol{q})\cdot\mathrm{d}\boldsymbol{q}\right)=\exp(i\pi\sigma_{x})=-\mathbb{1}$
(29)

In agreement with the computations in Sec. IV.

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