# Synthetic Kramers pair in phononic elastic plates and
helical edge states on a dislocation interface

Ting-Wei Liu  Fabio Semperlotti<EMAIL_ADDRESS>Ray W. Herrick
Laboratories, School of Mechanical Engineering, Purdue University, West
Lafayette, Indiana 47907, USA.

###### Abstract

In conventional theories, topological band properties are intrinsic
characteristics of the bulk material and do not depend on the choice of the
reference frame. In this scenario, the principle of bulk-edge correspondence
can be used to predict the existence of edge states between topologically
distinct materials. In this study, we propose and carefully examine a 2D
elastic phononic plate with a Kekulé-distorted honeycomb pattern engraved on
it. It is found that the pseudospin and the pseudospin-dependent Chern numbers
are not invariant properties, and the $\mathbb{Z}_{2}$ number is no longer a
sufficient indicator to examine the existence of the edge state. The
distinctive pseudospin texture and the pseudomagnetic field are also revealed.
Finally, we successfully devise and experimentally implement the synthetic
helical edge states on a dislocation interface connecting two subdomains with
bulk pattern identical up to a relative translation. The edge state is also
imaged via laser vibrometry.

topological phononics; synthetic Kramers pair; pseudospin; helical edge state;
dislocation interface

The study of topological states of matter has rapidly grown over the past few
decades [1, 2, 3]. While the field of topological materials originated and had
particular significance for the broader area of quantum mechanics, in recent
years these works served as an inspiration to explore the existence of analog
topological effects in classical waveguide systems [4, 5, 6, 7, 8]. On one
side, these analog quantum mechanical mechanisms offered engineers a powerful
route to design waveguides (either photonic, acoustic, or elastic) immune to
backscattering generated by disorder or defects. On the other side, these same
mechanisms were seen by physicists as ideal platforms to explore the effect of
design parameters and to experimentally validate important fundamental
concepts without the significant complexities imposed by the quantum scales.
In other terms, the artificial photonic and phononic structures were treated
as surrogate platforms to understand and develop the topological band theory
and its practical implications.

To-date, several analog mechanisms have been explored and validated for
classical waveguides. Initial attempts focused on the implementation and
transposition of the concept of quantum Hall effect. However, these systems
required the breaking of time reversal symmetry (TRS), which imposes
significant practical complexities due to the need for either special magneto-
optic and elastic materials, or for carefully controlled external input [9,
10, 11, 12, 13, 14, 15, 16, 17, 18]. More recently, mechanisms analog to TRS-
preserved quantum spin Hall effect (QSHE) [19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34] and quantum valley Hall effect [35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 31, 45, 46] were also proposed. These systems could be
built based on ordinary dielectric or linearly elastic materials, and only
required the breaking of spatial symmetry, which was a considerably more
practical approach. Other studies proposed the use of negative indices,
instead of artificial lattice structures, to realize analogues to the QSHE in
electromagnetic and acoustic systems [47, 48, 49, 50, 51, 52]. Among the TRS-
preserved mechanisms, bosonic systems can leverage the same valley degree of
freedom to realize an analog valley Hall effect. While the valley Hall effect
and the associated backscattering immunity of the edge states count on the
large separation of states in momentum space, the classical analog to QSH
systems relies on the decoupling of synthetic “pseudospin” states. These
states can potentially yield counter-propagating edge states that are close in
momentum space while still being robust against backscattering.

Despite these successful initial implementations, the emulation of a quantum
spin Hall topological insulator has proven to be a remarkably more complicated
task. It is well known that the $\pm\frac{1}{2}$-spin electronic systems
exhibit so-called Kramers pairs (given that the time-reversal operator
$\mathcal{T}$ applied twice returns the initial state with additional negative
sign, that is $\mathcal{T}^{2}=-\mathbf{1}$, where $\mathbf{1}$ denotes the
identity operator). It follows that under TRS, every eigenstate has a partner
with opposite momentum and spin and the same energy spectrum, that is
$E(\mathbf{k},\uparrow)=E(-\mathbf{k},\downarrow)$. This condition guarantees
the existence of gapless helical edge states that are time reversal
counterparts of each other with opposite spins. While both photons and phonons
are bosonic in nature, they do not possess the same intrinsic attributes.
Researchers have proposed different ideas to synthesize properties analog to
electron spins, often referred to as “pseudospins”. In phononic elastic
systems, examples include mixing of symmetric and antisymmetric Lamb modes in
waveguides [33, 34], and a “zone-folding” method [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32] to attain a doubly degenerate Dirac cone at the center of
the Brillouin zone that acts as a degree of freedom that resembles electron
spins. Nevertheless, it was reported that phononic systems exploiting this
mechanism could give rise to gapped edge states at zero momentum where the
$\omega$-$\mathbf{k}$ dispersion curves of the counter-propagating edge states
repel each other, due to coupling between them [28, 26, 27, 28, 29, 30, 21,
22, 23, 24, 25, 31]. These results showed that the edge states are not a
Kramers pair and do not have a continuous spectrum across the bulk band gap.
In addition, although the zone-folding approach is already widely adopted,
previous studies concentrated on mapping the system back to the electronic
counterpart but usually omitted explaining some discrepancies between the
synthetic phononic pseudospins and the electron’s intrinsic spin; hence,
leaving behind some obscure points such as the indeterminate pseudospin
states, and the seemingly indistinguishable topological phases.

In this paper, we propose an elastic analog of a topological insulator based
on a 2D phononic waveguide designed according to a Kekulé distorted honeycomb
pattern. The study uncovers interesting anomalous properties associated with
the generation of synthetic pseudospins and peculiar discrepancies with
electron spins. These anomalous properties include indeterminate pseudospin-
dependent Chern numbers and indistinguishable topological states, which are
characteristic of the “zone-folding” approach and different from electronic
QSH systems. Although the concepts of pseudospins and zone-folding have been
discussed in the literature, a series of questions were left unexplored. The
discrepancy between the classical “zone-folding system” and electronic QSH
systems leads to consider the former as an “imperfect analogue” since proper
“helical” edge state cannot be realized. In other terms, the counter-
propagating edge states are found to be gapped and with strong coupling at
$\mathbf{k}=0$. However, as we face the discrepancy and investigate its cause,
we identify a method to realize truly gapless helical edge states such that
the counter-propagating edge states are fully decoupled. In addition, we
leverage the unique ambiguity of the topological state to achieve edge states
along a dislocation interface connecting two sections of the same bulk
lattice. This latter condition is not achievable in QSH systems in which edge
states exist on an interface between materials exhibiting distinct and
determinate topological character.

This concept is also experimentally validated by directly imaging the response
of the phononic lattice via laser vibrometry, and by extracting the spectrum
of the edge state which matches well with the theoretical and numerical
predictions. It is worth to highlight that these findings are particularly
relevant from the practical perspective of a waveguide design. In fact, not
only this approach leads to robust and continuous gapless edge states, but it
also allows exploiting the concept of dislocation that greatly simplifies the
design. It is expected that the current design can be very well suited to
achieve vibration and structure-borne noise control capabilities fully
integrated in lightweight load bearing structures.

The proposed phononic lattice consists in an aluminum thin waveguide (i.e. a
thin plate) with a honeycomb-like groove pattern symmetrically engraved on
both sides. The groove has a variable thickness according to an extended
Kekulé distortion pattern. Fig. 1 (a) shows the benzene structure, originally
suggested by Kekulé, which contains alternating double and single bonds. We
adopt an extended version of the Kekulé distortion to define the groove depth
of the honeycomb cell which allows three different “bonds” (i.e. the
individual grooves forming the side of the cell) that can change their values
(i.e. their depths) continuously, as shown in Fig. 1 (b).

Figure 1:  (a) The benzene structure suggested by Kekulé which contains
alternating double and single bonds. (b) The extended Kekulé distorted pattern
used in our design. The bonds are replaced by the grooves (denoted in
different colors) having different and continuously varying depth (illustrated
by different line weight).

As shown in Fig. 2 (a), the phononic plate has thickness $b=1/4$ inch (6.35
mm). Before incorporating the Kekulé distortion, the honeycomb groove has a
constant depth $h_{0}$ and a lattice constant $a_{0}$. The Kekulé distortion
is then introduced by varying the groove depth according to the prescribed
pattern. Three different depths $h_{1,2,3}$ are assigned at the midpoints of
the hexagon edges, and the average depth $h_{0}=(h_{1}+h_{2}+h_{3})/3$ is
assigned at the vertices of the hexagons joining the edges. The grooves are
constructed by linearly connecting neighboring grooves, as schematically shown
in the isometric view in Fig. 2 (b). Figure 2 (b) also shows a top view of the
Wigner-Seitz unit cell in the inset. Note that the Kekulé perturbation
enlarges the primitive unit cell and the new lattice constant becomes
$a=\sqrt{3}a_{0}$, as shown in Fig. 2 (a). Also, the lattice symmetry is
degraded from $D_{6}$ to $D_{3}$ (or $C_{6v}$ to $C_{3v}$ in the context of 2D
lattices).

Figure 2:  (a) An illustration of the phononic elastic lattice consisting in
an aluminum plate with a honeycomb-like groove pattern symmetrically engraved
on both sides. Before the application of the Kekulé distortion, the lattice
has constant $a_{0}$. After the distortion is applied, the lattice constant
becomes $a=\sqrt{3}a_{0}$. (b) An isometric sectional view of a Wigner-Seitz
unit cell of a distorted lattice centered at a $C_{3}$ axis. Three different
depths $h_{1,2,3}$ are assigned to the midpoints of all the hexagon edges, and
the average depth $h_{0}=(h_{1}+h_{2}+h_{3})/3$ is assigned at the vertices
joining the edges. The grooves are constructed by linearly connecting
neighboring grooves. The inset shows the top view of the unit cell. (c) A
graphical representation showing the relation between the parameters
$(h_{0},\delta,\phi)$ and $(h_{1},h_{2},h_{3})$. (d) The parametric space to
generate all admissible lattices. The color coding should be interpreted as
follows: the hue indicates the parameter $\phi$ mod $2\pi/3$; the color
lightness indicates the distortion strength $1-|\delta|$. Different points
with the same color yield the same bulk lattice pattern up to a rigid body
translation (e.g., the three purple stars). Two examples of the irreducible
parametric domains to generate a unique bulk lattice pattern are shown as the
yellow dashed and the cyan dotted box. Depending on different paths, two
distinct bulk lattice can evolve from one to another with or without band gap
closing.

To facilitate the fabrication process, the groove width was chosen $D=1/16$
inch, dictated by the diameter of the (ball-end) machining tool. Also the
groove bottom was rounded so to match the actual slot cut by the tool.
Implementing these few considerations allowed a substantial ease of
fabrication via a computer numerical control (CNC) mill.

The three characteristic depths $h_{1,2,3}$ can be easily described by a set
of parameters $(h_{0},\delta,\phi)$

$\displaystyle h_{i}$ $\displaystyle=h_{0}+\delta_{i},$
$\displaystyle\delta_{i}$
$\displaystyle=\delta\cos\left(\phi+(i-1)\frac{2\pi}{3}\right),\quad i=1,2,3.$
(1)

This parameter set $(h_{0},\delta,\phi)\in\mathbb{R}^{3}$ can be considered as
the natural coordinate system of the parametric space $(h_{1},h_{2},h_{3})$
governing the band structure. The first parameter, the average depth $h_{0}$,
affects the overall frequency shift of the phononic band structure, and plays
no role in the topological transition. The second parameter $\delta$ controls
the magnitude of the Kekulé distortion and therefore is related to the opening
of the band gap. The parameter $\phi$ is the last free degree of freedom in
the 3D parameter space and, will be shown later, is at the root of the
peculiar behavior of the lattice. A graphical representation plotting the
relation between $(h_{0},\delta,\phi)$ and $(h_{1},h_{2},h_{3})$ is shown in
Fig. 2 (c). Note that given any real-number sequence $(h_{1},h_{2},h_{3})$,
one can always find an equivalent parameter set $(h_{0},\delta,\phi)$ that
yields $(h_{1},h_{2},h_{3})$. Given that the depth value must be positive, we
can restrict $h_{0}>0$ and $-1<\delta<1$.

Due to the properties of the cosine function, we have
$h_{i}|_{(\phi+\pi,\delta)}=h_{i}|_{(\phi,-\delta)}$, for all given
$(\phi,\delta)$. Thus, the domain of $(\phi,\delta)$ can be restricted to
$\left[0,\pi\right)\times(-1,1)$ while remaining capable of generating all
depth sequences $(h_{1},h_{2},h_{3})$, as shown in Fig. 2 (d). Another
interesting aspect is that given any three sequences $(h_{1},h_{2},h_{3})$,
$(h_{2},h_{3},h_{1})$ and $(h_{3},h_{1},h_{2})$ composed of the same three
numbers with the same cyclic permutation, they yield the same bulk lattice up
to some rigid body translation (or rotation) of the lattice (or, equivalently,
of the reference frame). For example, the permutation
$(h_{1},h_{2},h_{3})\rightarrow(h_{3},h_{1},h_{2})$ is identical to a
translation of the entire lattice by a vector $\left(0,-a_{0}\right)$ (see the
inset in Fig. 2 (b)), or a counterclockwise rotation of $2\pi/3$ with respect
to the point labeled $h_{0}$ along the $z$-axis in the inset of Fig. 2 (b). It
follows that the parameter space in Fig. 2 (d) contains three copies of a
certain subdomain that can generate the same bulk lattice. Two examples of the
irreducible subdomain are highlighted by the yellow dashed and the cyan dotted
boxes in 2 (d). One should find that $(\phi,\delta)$, $(\phi+\pi/3,-\delta)$
and $(\phi+2\pi/3,\delta)$ yield the same bulk lattice up to a rigid rotation
or translation, for all given $(\phi,\delta)$. Fig. 2 (d) shows all the
possible bulk lattice patterns identified by colors. Specifically, the color
reads $({\rm hue},{\rm lightness})=(\phi\mbox{ mod }2\pi/3,1-|\delta|)$ for
$\delta\geq 0$ and $((\phi+\pi)\mbox{ mod }2\pi/3,1-|\delta|)$ for $\delta<0$.
For example, the three purple stars in Fig. 2 (d) mark points of the parameter
space with the same color and, therefore, representing the same bulk lattice
(up to a rigid translation). This consideration leads to a peculiar result
that is, given a certain bulk lattice configuration (e.g. the red spot in Fig.
2 (d)), the lattice can evolve adiabatically into another (e.g. purple stars)
by following several different paths in the parametric space that may or may
not cross $\delta=0$. As previously mentioned (and clarified in detail in the
following), $\delta=0$ identifies the closing of the bulk band gap.
Considering that the closing and reopening of the band gap is an indicator of
a possible topological transition, the results in parameter space raises the
interesting question about the two configurations belonging or not to the same
topological phase. Along the same line of reasoning, one could wonder if the
two lattices belonging to the same bulk pattern and differing, at most, for a
relative rigid translation (or rotation) could belong to distinct
topologically phases; given that while crossing the interface between them the
band gap closes and reopens. If this latter question admitted a positive
answer, then edge states might exist at a dislocation interface between these
two phases. To gain more insight on this topic and answer the above question,
the phononic band structure and its topological properties must be analyzed in
detail.

Before introducing the Kekulé distortion (that is considering an unpertubed
lattice having $(\delta=0,h_{1}=h_{2}=h_{3}=h_{0})$), the “unit cell” in the
inset of Fig. 2 (b) is a supercell, and its original primitive lattice is
characterized by basis lattice vectors having $1/\sqrt{3}$ length and rotated
of $\pi/2$ with respect to those of the supercell. The band structure
associated with this supercell exhibits a double Dirac cone at the $\Gamma$
point (Fig. 3 (b)) that is the result of the folding of the two cones at the
valleys ${\rm K}_{0}(\frac{2\pi}{\sqrt{3}a_{0}},\frac{-2\pi}{3a_{0}})$ and
${\rm K}^{\prime}_{0}(\frac{2\pi}{\sqrt{3}a_{0}},\frac{2\pi}{3a_{0}})$ of the
original primitive cell. This concept is visually exemplified in Fig. 3 (a)
where the solid black hexagon indicates the first Brillouin zone (BZ)
associated with the supercell and the black dashed hexagon shows the first BZ
of the original primitive cell. The gray hexagons indicate the
$\mathbf{k}$-space duplicates of the BZs. The original Dirac degeneracies at
the valleys ${\rm K}_{0}$ and ${\rm K}^{\prime}_{0}$ of the primitive cell
were a consequence of intact space inversion symmetry (SIS) and TRS, while the
new four-fold degeneracy at the $\Gamma$ point of the supercell is an artifact
of band folding and merely due to the artificial selection of the supercell.
However, the supercell could be taken as the new reference configuration (i.e.
the primitive unit cell) when a local perturbation (such as the Kekulé
distortion) is applied to it. The symmetry breaking resulting from the local
perturbation would lift the four-fold degeneracy opening a band gap at the
$\Gamma$ point.

Figs. 3 (b) and (c) show the phononic band structures of the reference lattice
($\delta=0$) and a distorted one ($\delta=0.5,\phi=0$), respectively. The band
structure was obtained by numerically solving the elastodynamic governing
equations for the Bloch eigenstates. The black curves in the plots correspond
to antisymmetric guided Lamb ($A$) modes, which are essentially flexural modes
of the plate waveguide. The light gray curves show the symmetric ($S$) modes.
Given the phononic plate is symmetric with respect to its neutral plane, the
flexural modes are completely decoupled from the symmetric modes and from now
on we will focus only on the flexural modes.

In Fig. 3 (b), the double Dirac cone with four-fold degeneracy shows at the
$\Gamma$ point, and the frequency bands are doubly degenerate throughout
$\Gamma$-K. However, they splits along the $\Gamma$-M direction. This is due
to the anisotropy of the band structure near the valleys ${\rm K}_{0}$ and
${\rm K}^{\prime}_{0}$. Indeed, it is known that the dispersion around the
valleys is only isotropic under linear $\mathbf{k}$ approximation, and shows
trigonal warping away from the valleys [53]. The two mirrored trigonally
warped cones cross each other. The frequency spectra remain degenerate along
the $\Gamma$-K section, corresponding to the original ${\rm K}_{0}$-${\rm
K}_{0}$ and ${\rm K}^{\prime}_{0}$-${\rm K}^{\prime}_{0}$ directions (which
happens to be the intersecting part of the two warped cones), while split
along the $\Gamma$-M direction that corresponds to the original ${\rm
K}_{0}$-$\Gamma$ and ${\rm K}^{\prime}_{0}$-${\rm K}_{0}$ directions (see Sec.
IV and Fig. 11 in the Supporting Information (SI) for a drawing helping the
visualization of this band structure). This observation already indicates a
major discrepancy between the phononic “zone-folding” system and the
electronic 2D topological insulators. For the latter, in the absence of
inversion asymmetry, and spin-orbital coupling, the entire energy band
structure are doubly degenerate. On the contrary, for phononic systems, the
two-fold degeneracy only occurs along six discrete directions
($\Gamma\mbox{-K}$ and $\Gamma\mbox{-K}^{\prime}$). Note that, although these
lines of intersection connected at the $\Gamma$ point might be reminiscent of
exceptional points, the degeneracy at the $\Gamma$ point is still a diabolic
point given that the system is Hermitian [54].

Figure 3:  (a) Schematic of the reciprocal space illustrating the formation of
the double Dirac cone by zone folding. The solid black hexagon indicates the
first Brillouin zone (BZ) associated with the supercell and the black dashed
hexagon shows the first BZ related to the original primitive cell. The Dirac
cones at the two valleys ${\rm
K}_{0}(\frac{2\pi}{\sqrt{3}a_{0}},\frac{-2\pi}{3a_{0}})$ and ${\rm
K}^{\prime}_{0}(\frac{2\pi}{\sqrt{3}a_{0}},\frac{2\pi}{3a_{0}})$ of the
primitive cell move to and overlap at the $\Gamma$ point of the supercell. (b)
The phononic band structures of the reference lattice ($\delta=0$) and (c) of
a Kekulé distorted lattice with $\delta=0.5,\phi=0$.

In Fig. 3 (c), under the effect of the Kekulé distortion, the four-fold
degeneracy is lifted and a band gap opens up, as indicated by the pink shaded
box. On the other hand, the spectra along $\Gamma$-K, are now lifted with the
only exception of an isolated two-fold degeneracy at the $\Gamma$ point. We
anticipate that this behavior could be interpreted as an effective pseudospin-
orbit coupling that splits the two pseudospin bands. We should also note that,
in electronic systems, even in presence of spin-orbit coupling, the Kramers
theorem requires that double degeneracies occur both at the $\Gamma$ and the
three M points (given that the M points are also invariant under
$\mathbf{k}\rightarrow-\mathbf{k}$ as they are midpoints between two
neighboring $\Gamma$ points) [55]. Obviously the “zone-folding” system does
not follow the same behavior, and shows significant splitting at the M point
(Fig. 3 (b) and (c)). This again points out the discrepancy between the two
systems.

In fact, these substantial differences can be ascribed to different symmetry
operators that protect the degeneracy at the $\Gamma$ point. In electronic QSH
systems, time-reversal symmetry and $\mathcal{T}^{2}=-\mathbf{1}$ guarantee
the Kramers degeneracy at time-reversal invariant points such as the $\Gamma$
and M points in the $\mathbf{k}$-space. On the other hand, in the current
elastic lattice, we show that only the following operators property
$(C_{3}\mathcal{T}_{F})^{2}=-1C_{3}^{2}$ holds in the vicinity of the $\Gamma$
point, where the operator $\mathcal{T}_{F}$ takes the expression of the time-
reversal operator for spin-$\frac{1}{2}$ fermionic systems. The antiunitary
operator $C_{3}\mathcal{T}_{F}$ maps the system between different azimuth
angles in the $\mathbf{k}$-space (i.e., the angle between the $k_{x}$-axis and
the current $\mathbf{k}$-vector), and the $\Gamma$ point becomes the
$(C_{3}\mathcal{T}_{F})$-invariant self-dual point [56], which leads to the
degeneracy at the $\Gamma$ point. See further details on this point in Sec.
III in SI.

This crucial difference ultimately affects how the topological index of a bulk
lattice can be defined. Similarly to the TKNN number [57] that cannot classify
nontrivial electronic QSH systems from ordinary ones, the typical
$\mathbb{Z}_{2}$ index for electronic QSH system also becomes inappropriate
for the current system (since it is based on $\mathcal{T}^{2}=-\mathbf{1}$).
Instead, we define a local topological order that resembles the
$\mathbb{Z}_{2}$ index in that it is based on integrals of the pseudospin-
resolved Berry curvature localized near $\Gamma$ (pseudospin-Chern number), as
described in Sec. VI. The reader might doubt the legitimacy of this approach
under the scenario of pseudospin coupling. We show in Sec. IX of SI that the
even in the presence of pseudospin coupling, the off-diagonal elements of the
Chern number matrix vanish due to odd $\mathbf{k}$-space symmetry of the
inter-pseudospin Berry curvature and the result is the same as when the
pseudospins decouple.

One of the standard procedures to analyze the system in proximity of the
degeneracy is through the $\mathbf{k}\cdot\mathbf{p}$ approach (or similar
low-order perturbation approaches), which expands the eigenstates around the
$\Gamma$ point by using a basis formed by the degenerate eigenstates. This
procedure leads to a description of the system in the form of a $4\times 4$
block diagonal Hamiltonian that is quadratic in $\mathbf{k}$. After the
application of a proper change in basis [26, 30], this Hamiltonian can be
mapped to the Bernevig-Hughes-Zhang (BHZ) model of topological insulators.
This simplified Hamiltonian yields four frequency bands that are quadratic in
$\mathbf{k}$ with the two upper and the two lower bands being doubly
degenerate, respectively. The basis used to obtain this form of the
Hamiltonian defines the pseudospin eigenstates.

Typically, the perturbed lattices are classified as “nontrivially” gapped
materials if they satisfy either one of the following criteria: 1) the spectra
of the $p$\- and $d$-orbitals are inverted [21], or 2) $B/M<0$ (where $B$ and
$M$ are parameters from the BHZ Hamiltonian) [22]. However, it can be seen
(details provided in SI) that in a continuous and periodic medium, the parity
of the wavefunctions used in classifying $p$\- or $d$-orbitals depends on the
reference frame. It is also reported that the mapping to the BHZ model
actually depends on the gauge choice or the unit cell selection [26, 30],
therefore the possible “nontrivial” character of the lattice is indeterminate.
On the other hand, unlike nontrivial 2D topological insulators that support
gapless helical edge states on the boundary, the phononic lattices classified
as “nontrivial” still do not support edge states on its boundary with either
vacuum or air. In these materials, edge states only exist on the interface
between two gapped lattices. Therefore, the classification as a nontrivial
material acquires a somewhat more arbitrary character and the conventional
bulk-edge correspondence principle does not apply.

In addition, the low-order Hamiltonian cannot capture the band splitting along
the $\Gamma$-M direction which is a distinctive feature compared with
electronic systems. The band splitting of pseudospin pairs along the
$\Gamma$-K direction occurs due to the higher order pseudospin-orbit coupling
terms. These arguments show that the conventional perturbation approach is not
sufficient for an accurate description of the phononic Kekulé lattice.

We also carry out a thorough analysis of the topological band properties using
ab initio calculations (details provided in SI). An explicit expression for
the phononic pseudospins is provided and numerically evaluated based on the
circulation integral of the acoustic Poynting vector throughout the unit cell
that tiles periodically covering the entire real $\mathbf{x}$-space. This
concept bears analogy with the $\mathbf{k}$-space representation of the
acoustic spin and orbital angular momenta discussed in the literature [58, 59,
60] (see Sec. IV in SI) .

Results show that the pseudospin indices actually depend on the choice of the
reference frame, and consequently the pseudospin-dependent Chern numbers and
the local topological order are also non-unique. This observation opens the
way to conceive gapless edge states on the interface between lattices having
the same bulk pattern but differing up to a relative translation, as
illustrated in the following section.

Electronic topological insulators have a pair of helical edge states that are
topologically protected, counter-propagating, and with gapless energy spectra
across the entire band gap. One of the key element leading to the gapless
condition is the Kramers degeneracy guaranteed at $\mathcal{T}$-invariant
points in $k_{\parallel}$-space including $k_{\parallel}=0$. On the other
hand, the existing literature shows that, in a phononic analog of a
topological insulator based on the zone-folding approach, the edge states are
generally gapped at $k_{\parallel}=0$. This result is due to the fact that
Kramers theorem is valid only for systems with half-integer total spin, such
as electronic systems. In phononic systems, the“synthetic Kramers pair” is
created by the pseudospins. When the two pseudospin states are close (in
$\mathbf{k}$-space), they tend to repel each other and the eigenstates show
strong mixing that completely annihilates the pseudospins. Given that the
pseudospin is generally not conserved, the Kramers degeneracy does not hold
and the gap at $k_{\parallel}=0$ typically appears for the phononic edge
states. This observation also means that the edge states are not robust and
prone to backscattering since the counter-propagating edge states are already
strongly coupled and gapped even before introducing defects. Previous studies
ascribed the presence of the gap to the lack of $C_{6v}$ symmetry of the
lattice near the domain wall interface [21], and attempted to reduce the gap
width by smoothing the transition between the two lattices [24]. However, as
we find out in the following, the gapless edge state does not strictly require
local $C_{6v}$ symmetry and can exist on an abrupt dislocation interface.

Given the pseudospins have a certain distributed pattern throughout the
phononic lattice, the position of the edge which terminates the bulk lattice
(or equivalently, the translation of the bulk pattern relative to the edge) is
certainly expected to serve as a key parameter and to affect the behavior of
the edge state propagating on it. A proper skipping-orbit condition on the
edge can in fact realize nonrestrictive ballistic transport of the phonons and
therefore, achieving fully decoupled, counter-propagating, and gapless
phononic helical edge states. In a recent study [30], it was shown that this
condition can be realized by fine-tuning the parameter $\phi$ of the two
adjacent lattices that are space-inverted images of each other. In the present
study, in order to demonstrate that the edge state can exist on a dislocation
interface, we let the adjacent lattices have the same bulk lattice pattern so
that they can differ only up to a relative translation of amplitude $a_{0}$
along the $y$-direction. Such translation distance preserves the overall
honeycomb pattern across the two lattices which, although not necessary, makes
the fabrication of the material sample very easy. Under a fixed reference
frame, this lattice assembly can also be considered as integrating two
lattices with parameters $\phi$ and $\phi+2\pi/3$. To make it even simpler, we
let $\phi=0$, so that the dislocation interface is a plane of mirror symmetry.
Once again, these assumptions do not limit the generality of the results but
facilitate the final lattice configuration to be fabricated and used for the
experiment. Indeed, under these conditions, the edge state will be symmetric
with respect to the interface, hence also facilitating its excitation via a
single transducer placed right on the interface.

By modifying the depth of the groove located on the interface, the edge state
dispersion can be readily tuned. In particular, by sweeping through different
depths, we find the value such that the edge states reaches accidental
degeneracy at $k_{\parallel}=0$ (see SI for details). Fig. 4 (a) shows the
edge state spectrum along the optimized dislocation interface, and Fig. 4 (b)
shows the strain energy distribution and the mechanical energy flux of the
eigenstate indicated by the arrow in Fig. 4 (a). The color in Fig. 4 (a)
denotes the pseudoangular momentum of each eigenstate, whose integral is
calculated over a complete hexagonal cell located just next to the
dislocation, as shown in the dashed cyan hexagon in Fig. 4 (b).

Figure 4:  (a) The edge state spectrum along the optimized dislocation
interface. The color denotes the pseudoangular momentum. (b) The strain energy
distribution (color) and the mechanical energy flux (yellow arrow, log scale)
of the eigenstate indicated by the arrow in (a). The dashed cyan hexagon shows
the domain used for the pseudoangular momentum integral. The energy flux near
the interface indicates a typical skipping-orbit pattern.

These results also show that the two edge states are indeed pseudospin
polarized, fully decoupled, counter-propagating and gapless, therefore a
successful realization of a synthetic Kramers pair. The mechanical energy flux
near the symmetric dislocation interface in Fig. 4 (b) also indicates a
typical skipping-orbit pattern. Note that the edge state is achieved on an
abrupt dislocation, that is without a transition zone smoothing the change in
the lattice pattern; in addition, the unit cells on the dislocation do not
possess $C_{6v}$ symmetry.

In order to experimentally validate this concept of topological elastic
material, we fabricated the phononic plate having two subdomains made of the
same bulk honeycomb pattern but differing up to a relative translation. The
two domains were used to define a Z-shape dislocation interface (Fig. 5)
similarly to the example illustrated in the previous section. Further details
on the fabrication and the experimental setup are provided in SI).

Figure 5:  (a) An image of the fabricated phononic plate. The Z-shape
dislocation interface between the two subdomains is indicated by the white
dashed line. Piezoelectric ceramic transducers were glued at the top terminal
of the Z-dislocation and on both sides of the plate, in order to generate
$A_{0}$ Lamb waves. (b) An illustration of the transducer system and its
connections. The red arrows indicate the poling direction of the piezoelectric
ceramic plates. (c) The measured velocity amplitude at 172 kHz. The white
dashed box includes the data points selected for 1D Fourier transform. (d) The
$k_{x}$-spectra in the range 150-200 kHz. The calculated pseudospin-up edge
state (cyan dashed curves) and bulk band (white dotted curve) dispersion
curves are superimposed to the experimental results. The curves were down
shifted by 7 kHz (fractional shift of $-3.8\%$) to match with the experiment
data. Results confirm that the edge state along the dislocation is indeed
gapless and there is no evidence of coupling to the counter-propagating edge
state due to the presence of sharp corners on the path.

Two piezoelectric ceramic patches were glued at the top terminal of the
Z-dislocation, on both sides of the plate, as shown in Fig. 5 (a, b). The red
arrows in Fig. 5 (b) indicate the poling direction of the piezoelectric
ceramic plates. They were aligned in the same $+z$-direction so to act out-of-
phase when applying the voltage on their outer electrodes. This configuration
helps an efficient excitation of (primarily) the $A_{0}$ mode. The response of
the plate in terms of the velocity field distribution was measured by a
scanning laser Doppler vibrometer.

Figure 5 (c) shows the measured velocity amplitude at 172 kHz, which indicates
the edge state propagating along the Z-dislocation. To obtain the edge state
dispersion from the measured data, data points along the first segment of the
Z-dislocation (enclosed in the white box in Fig. 5 (c)) were selected and
Fourier transformed so to obtain the $k_{\parallel}$-spectra (in the range
150-200 kHz) shown in Fig. 5 (d). The calculated pseudospin-up edge state
(cyan dashed curves) and the bulk band dispersion curves (white dotted curve),
previously shown in Fig. 4(a), are superimposed on the spectrum. The curves
were downshifted by 7 kHz (fractional shift of $-3.8\%$, normalized by the
calculated edge state center frequency) to align with the experimental data.
This is a small error considering the operating frequency range and it is
likely due to slight deviations of the mechanical properties of the aluminum
alloy from the nominal values as well as to small fabrication imperfections.
Nevertheless, results confirm that the edge state along the dislocation is
indeed gapless across the bulk band gap and that there is no evidence of
coupling to the counter-propagating edge state in the presence of sharp
corners on the path.

The weak attenuation visible along the consecutive sections of the interface
is likely the result of a few factors. a) There is energy accumulation on the
top edge due to the limited absorbing performance of the viscoelastic layer.
This accumulation acts as an incoherent source of energy that feeds back into
the channel. After this incoherent component of the input is reflected at the
first corner, the two remaining branches appear markedly more uniform. b) The
measured response contains not only the edge state, but also the evanescent
bulk wave in the band gap. The response of the latter decays exponentially
(anisotropically) from the transducers, therefore the mixed response shows
larger amplitude on the first section. c) The decay of the edge state is
partially due to the asymmetry of the phononic plate. In the SI, we describe
some inconsistencies in the machining results on the two sides which leads to
a slight asymmetry with respect to the mid-plane. The asymmetry caused weak
coupling between antisymmetric and symmetric modes. The continuous spectrum of
the symmetric modes in the bulk band gap of the antisymmetric modes leads to
some energy leakage into the bulk symmetric modes as a result of the weak
coupling between $A$ and $S$ modes. We also conducted a time-transient
measurement using a wave packet centered at 172 kHz as the input signal
showing that the wave packet propagates through the two acute corners without
observable reflections. A full video of the transient response is also
provided in SI [61].

This study presented a fully continuous 2D elastic topological waveguide based
on the concept of a Kekulé distorted lattice and capable of achieving truly
gapless phononic edge states. A combination of theoretical and numerical
results showed that this phononic pseudospin Hall system cannot be studied and
entirely understood by means of traditional perturbation techniques. Reverting
to an approach based on in-depth numerical simulations allowed us to show a
discrepancy of the degenerate band structure of “zone-folding” systems
compared to electronic quantum spin Hall systems. This approach also yielded
the fine structure of the spectrum splitting due to pseudospin-orbit coupling
caused by inversion asymmetry, also not observable via standard perturbation
techniques. We introduced a definition of the pseudospin that allowed all
eigenstates in $\mathbf{k}$-space to be labeled clearly and that ultimately
revealed a distinctive alternating six-lobe pseudospin texture. By direct
observation of the eigenstates, we also isolated the origin of the ambiguity
of the pseudospin state, and constructed the $\mathbf{r}$\- and
$\mathbf{k}$-dependence of the pseudomagnetic field associated with the
symmetry breaking. The ambiguity in the pseudospin state also leads to
indeterminate pseudospin dependent Chern numbers. We find that the Kekulé
lattices have the local topological order $\nu=\pm 1/2$ which suggests the
possible existence of edge states. While phononic gapless edge states at zero
momentum had not been observed previously, we showed numerical and
experimental evidence of their existence in our elastic phononic system. More
specifically, edge states were found to be able to propagate on a dislocation
interface of two adjacent identical bulk lattices differing only up to a
relative translation. The robustness of these states was examined by
propagating elastic wave packets into the dislocation waveguide with no
observable backscattering at sharp corners along the path. The experimental
results were fully consistent with the dispersion structure predicting gapless
edge states and decoupled pseudospin polarizations, therefore suggesting that
the two counter-propagating edge states form a successful synthetic Kramers
pair.

The approach presented in this study suggests a simple yet robust approach to
turn any elastic plate structure into a topological waveguide capable of
backscattering protected states while still preserving its structural
properties. The possibility of creating such highly controlled states on
simple dislocations also greatly simplify both design and fabrication. It is
possible to envision the application of this concept to customize the dynamic
response of thin wall structures in the context of, as an example, vibration
and structure-born noise control, acoustic signal transmission, analog wave
filtering.

ACKNOWLEDGMENTS

The following work was partially supported by the Office of Naval Research
under grant #N00014-20-1-2608, and by the National Science Foundation (NSF)
under grant #1761423.

SUPPORTING INFORMATION

###### Contents

  1. I Ambiguity in p/d\- orbital classification
  2. II Calculation of the band structure
  3. III Symmetry protected degeneracy and its connection to the Kramers degeneracy
  4. IV Pseudospins
  5. V Berry curvature and pseudomagnetic field
  6. VI Local topological order
  7. VII Pseudospin texture for a general case
  8. VIII Numerical method for Berry curvature evaluation
  9. IX Chern number matrix
  10. X Topology of the $(\phi,\delta)$ parametric space
  11. XI Dislocation interface to achieve decoupled gapless edge states
  12. XII Fabrication and experiment setup

##  I Ambiguity in p/d\- orbital classification

In the context of “zone-folding” systems, [21] adopted a criterion to
distinguish “nontrivial” from “ordinary” materials based on the inversion of
$p$\- and $d$-bands. However, we note that there is an intrinsic ambiguity in
the classification of $p$\- and $d$-orbitals. Figure 6 (a) shows an example of
a periodic wavefunction where red and blue colors indicate opposite signs of
the wavefunction.

Figure 6:  (a) A periodic wavefunction where red and blue color indicate
opposite signs of the wavefunction. The solid and dashed hexagons show two
possible choices of the Wigner-Seitz cell. (b) With the solid hexagon chosen
as the unit cell, the wavefunction is symmetric with respect to the $x$-axis
and antisymmetric with respect to the $y$-axis, therefore the wavefunction has
odd parity with respect to the unit cell center, and shows a $p_{x}$-orbital
pattern. (c) When the dashed hexagon is taken as the unit cell, both $x$\- and
$y$-axes are planes of antisymmetry, and the wavefunction has even parity. The
four-lobe alternating pattern of the wavefunction is characterized as a
$d_{xy}$-orbital.

For a continuous periodic lattice such as the considered phononic plate, there
is no restriction in choosing the “atomic site” as center of the unit cell. In
Fig. 6 (a), the solid and dashed hexagons show two possible choices of the
Wigner-Seitz cell. When taking the solid hexagon, the wavefunction is
symmetric with respect to the $x$-axis and antisymmetric with respect to the
$y$-axis, therefore the wavefunction has odd parity with respect to the unit
cell center, and appears as a $p_{x}$-orbital-like wavefunction (see Fig. 6
(b)). At the same time, by taking the dashed hexagon as the unit cell, both
$x$\- and $y$-axes are planes of antisymmetry, and the wavefunction has even
parity. The four-lobe alternating pattern of the wavefunction is also
characterized as the $d_{xy}$-orbital, as shown in Fig. 6 (c). Moreover, the
general Kekulé distorted lattice usually has broken inversion symmetry, so
strictly speaking the wavefunctions always have mixed parities.

Such ambiguity does not make this criterion always applicable. Together with
the gauge ambiguity in mapping to the BHZ model [26], it appears to be
meaningless to classify the topological nature of these phononic lattices
following the standard approach for electronic topological insulators.

##  II Calculation of the band structure

The band structure is obtained by parametrically solving for $\omega$ while
varying $\mathbf{k}$ around the irreducible BZ. $\omega$ is obtained as the
eigenvalue from the elastodynamic boundary value problem:

$\displaystyle(\lambda+2\mu)\nabla(\nabla\cdot\mathbf{u})-\mu\nabla\times(\nabla\times\mathbf{u})=-\rho\omega^{2}\mathbf{u},$
(2) $\displaystyle\mathbf{u}_{\rm destination}=\mathbf{u}_{\rm
source}e^{-i\mathbf{k}\cdot(\mathbf{r}_{\rm destination}-\mathbf{r}_{\rm
source})},$ (3)

where Eq. 2 is the Navier’s equation governing the displacement field
$\mathbf{u}(\mathbf{r})$ in the elastic domain, $(\lambda,\mu)$ are the
elastic Lamé constants, and $\rho$ is the mass density. We assume
$(\lambda,\mu)=(50.4,25.9){\rm GPa}$ and $\rho=2700{\rm kg}/{\rm m}^{3}$
corresponding to an aluminum alloy 6063-T83 which has similar composition to
that of the Mic6${}^{\mbox{\textregistered}}$ aluminum cast plate used in the
experiment. Eq. 3 is the boundary condition applied on each of the three pairs
of opposite vertical boundaries shown in Fig. 2 (b) in the main text to solve
for the Bloch eigenstates. The equations are solved using the commercial
finite element package COMSOL Multiphysics.

Fig. 7 shows the band structures for lattices having $\phi=0,\pi/6,\pi/3$,
respectively ($\delta\equiv 0.5$).

Figure 7:  The band structures for lattices having (a) $\phi=0$, (b)
$\phi=\pi/6$, and (c) $\phi=\pi/3$, respectively. In all three cases
$\delta=0.5$.

In the special cases $\phi=0~{}{\rm or}~{}\pi/3+2n\pi/3,\forall
n\in\mathbb{N}$ (Fig. 7 (a,c)), the inversion symmetry is intact, the lattice
still possesses $C_{6v}$ symmetry (though the number of $C_{6}$ axes is
reduced to $1/3$ compared with the reference lattice and the original
$a_{0}$-periodicity is also expanded to $a=\sqrt{3}a_{0}$), and deterministic
degeneracies show at the K point. Fig. 7 (b) with $\phi=\pi/6$ represents a
more general case in which the lattice symmetry is lowered to $C_{3v}$ and the
inversion symmetry is broken. As a result, the degeneracies at the K point are
lifted. We note that, not only the two upper (lower) bands forming the double
cone are degenerate at the $\Gamma$ point, but there are also degeneracies
involving other bands in the same frequency range that are inseparable from
the currently studied double cone system. For example, in Fig. 7 (b,c) along
the $\Gamma$-M section, a third frequency band crosses one of the frequency
band of the upper cone.

Figure 8:  A close up view of the upper cone of the $\phi=\pi/6$ band
structure. The seemingly independent spectra are actually smoothly connected
at the K point.

Fig. 8 shows a zoom-in view of the upper cone of the $\phi=\pi/6$ band
structure. In addition to the two bands forming the upper cone (the blue solid
and the red dotted curves), there is another band around the same frequency
range (the green dashed curve). Due to the broken inversion symmetry, there is
no degeneracy at the K point; yet, these seemingly independent bands are
smoothly connected at the K point (one can connect the left to the right in
Fig. 8, hence forming a loop) forming one continuous closed curve. In other
words, although the band structure shows three curves that are apparently
independent, all states on these three bands can evolve continuously from one
to another. A rigorous analysis approach would involve a non-Abelian Berry
phase that accounts for interactions between these three bands. Also, the
lifting of the degeneracy induced by the inversion-symmetry-breaking is also
expected to contribute additional geometric phase around the K point as was
shown in valley Hall systems [35, 36, 37, 38, 40, 43]. These effects
complicate the evaluation of the topological order of the entire band
(requiring the integral of the Berry curvature throughout the BZ). We
anticipate that, similar to the quantum (acoustic) valley Hall materials, the
dynamic behavior (especially the edge state) of the lattice relies on the
local geometric phase around the double cone, therefore we can focus the
analysis on the neighborhood of the $\Gamma$ point.

##  III Symmetry protected degeneracy and its connection to the Kramers
degeneracy

Figure 7 clearly illustrated that the use of a general Kekulé distortion would
result in the formation of two two-fold degeneracies at the $\Gamma$ point
that are left behind after lifting the four-fold Dirac degeneracy of the
initial unperturbed lattice. In this section we show that the degeneracy can
be connected to the existence of a space-time symmetry operator, whose meaning
is closely connected to the concept of Kramers degeneracy in electronic
systems.

Recall that, for systems with half-integer total spin, it is always true that
$\mathcal{T}^{2}=-\mathbf{1}$, where $\mathbf{1}$ stands for the identity
operator. Whenever the system possesses time-reversal symmetry, the system is
at least two-fold degenerate, which is known as Kramers degeneracy [62]. The
degeneracy is guaranteed by three conditions: i)
$\mathcal{T}^{2}=-\mathbf{1}$, ii) the operator $\mathcal{T}$ is antiunitary
[63], and iii) the time-reversal symmetry of the system is preserved,
$\mathcal{THT}^{-1}=\mathcal{H}$ where $\mathcal{H}$ denotes the system
Hamiltonian.

As discussed in the main text, $\mathcal{T}^{2}$ is always $+\mathbf{1}$ for
the present elastic lattice, which means that the Kramers theorem does not
apply in a strict sense. However, if an antiunitary operator $\mathcal{A}$
(instead of $\mathcal{T}$) exists and satisfies all the three conditions
mentioned above, considerations that are conceptually analogous to the Kramers
theorem can be drawn and will still assure the existence of a two-fold
degeneracy; this concept was recently demonstrated by Fruchart et al. [56].
Specifically, in our system, the operator $\mathcal{A}=C_{3}\mathcal{T}_{F}$,
where $C_{3}$ is an operator corresponding to a proper clockwise rotation in
increments of $2\pi/3$ and $\mathcal{T}_{F}$ is the fermionic time-reversal
operator. At the $\Gamma$ point, the operator $\mathcal{A}$ is found to
satisfy the same three conditions dictated by the Kramers theorem if the first
condition is slightly modified. Indeed, $\mathcal{A}$ satisfies:

  1. 1.

$\mathcal{A}^{2}=(C_{3}\mathcal{T}_{F})^{2}=-C_{3}^{2}\mathbf{1}$.

  2. 2.

$\mathcal{A}$ is antiunitary.

  3. 3.

$\mathcal{AHA}^{-1}=\mathcal{H}$.

In the following, we show that these three conditions hold and, although
representing a modified version of the original Kramers theorem, they still
assure the degeneracy at the $\Gamma$ point.

Assume that in the vicinity of the $\Gamma$ point, the system dynamics of the
four bands (the lifted double cones) can be described by a $4\times 4$
Hamiltonian. Although we stated that the low order approximation cannot
describe the system dynamics accurately away from the $\Gamma$ point, here we
are interested in the degenerate property at the $\Gamma$ point, and the
$4\times 4$ Hamiltonian model is again a convenient tool.

At this stage, we do not know if a degeneracy exists at the $\Gamma$ point for
the Kekulé distorted lattice. We also do not know the system Hamiltonian
around the $\Gamma$ point. Although we can obtain the numerical values of the
$4\times 4$ matrix elements by selecting a basis, such as the four degenerate
eigenstates of the unperturbed lattice, then expand the perturbed eigenstates
and back up the Hamiltonian (see Supplemental Material of [30]). Instead,
simply by examining the symmetry of the eigenstates $\Ket{\mathbf{n}}$ in the
neighborhood of the $\Gamma$ point (which they are obtained from the first-
principle numerical calculation), we can easily show the three conditions for
$\mathcal{A}$ hold.

To be more specific, let us examine the four eigenstates in the vicinity of
the $\Gamma$ point along the $\pi/6$ azimuth angle in $\mathbf{k}$-space
($\lVert\mathbf{k}\rVert=10\mathrm{m}^{-1}=0.035\Gamma\mbox{-M}$) as shown in
the leftmost column in Fig. 9. The four eigenstates
$\Ket{\mathbf{n}},n=1,2,3,4$ show the “periodic parts” of the $n$th Bloch
states, on their mid-plane $z$-displacement.

Figure 9:  The consecutive application of the antiunitary operator
$\mathcal{A}=C_{3}U_{F}K$ to the eigenstates at
$\|\mathbf{k}\|=10,\angle\mathbf{k}=\pi/6$ (0.035$\Gamma$-M). A single
application of the operator maps the original set of eigenstates to those
corresponding to the system at $\angle\mathbf{k}=\pi$. A second application of
the same operator maps them to the eigenstates corresponding to the system at
$\angle\mathbf{k}=-\pi/6$, that is the eigenstates correspond to the original
set rotated counterclockwise by $2\pi/3$ and with an additional negative sign.
The wavefunctions shown here are the periodic part of the Bloch state of
$z$-displacement on the min-plane of the plate, in a Wigner-Seitz cell.

A lattice with general Kekulé distortion possesses $C_{3v}$ symmetry. Taking
$\mathbf{k}$ parallel to one of the three planes of mirror symmetry
($\angle\mathbf{k}=\pi/6$), the system Hamiltonian at the specific
$\mathbf{k}$ point also inherits that specific mirror symmetry. It follows
that the eigenstates are either symmetric or antisymmetric about the symmetry
plane, as shown by the numerical results in the first column of Fig. 9. On the
other hand, if the propagation vector $\mathbf{k}$ is perpendicular to a
symmetry plane (for example, along $\Gamma\mbox{-K or K}^{\prime}$,
$\angle\mathbf{k}=0,\pi$), the eigenstates also feature the same
(anti-)symmetry about the mirror plane when $\lVert\mathbf{k}\rVert$
approaches 0, (examples include the well-known acoustic $(1,1)^{T}$ and
optical $(1,-1)^{T}$ branches of a diatomic phonon lattice as $k\rightarrow 0$
[64]). The numerical results shown in the 4th and 6th columns in Fig.9 again
highlight the symmetry of the eigenstates.

We can define explicitly the operator $\mathcal{A}=C_{3}\mathcal{T}_{F}$ to be
applied to a wavefunction
$\Ket{\psi}=c_{n}\Ket{\mathbf{n}_{\angle\mathbf{k}=\pi/6}}\dot{=}(c_{1},c_{2},c_{3},c_{4})^{T}$,
under $\\{\Ket{\mathbf{n}_{\angle\mathbf{k}=\pi/6}}\\}$-basis representation.
To get the expression of $\mathcal{T}_{F}$, we first recall that the time
reversal operator is an antiunitary operator $\mathcal{T}=UK$, where $U$ is a
unitary operator and $K$ is the complex conjugate operator. For spinless
systems, such as our classical elastic system, it is simply $\mathcal{T}=K$.
For spin-$\frac{1}{2}$ fermionic systems, $U=e^{i\theta}\sigma_{y}$ (under
$z$-spin representation) where $\theta$ is an arbitrary phase and $\sigma_{y}$
is the Pauli $y$ matrix [63]. Let $\theta=\pi/2$,

$U=e^{i\pi/2}\sigma_{y}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$ (4)

By replacing the $z$-spin basis by each (symmetric and antisymmetric) pair of
our current $\\{\Ket{\mathbf{n}}\\}$-basis, we can obtain the analogue
fermionic time reversal operator for our classical mechanical system,
$\mathcal{T}_{F}=U_{F}K$, where $U_{F}$ is a block-diagonal matrix,

$U_{F}=\begin{pmatrix}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\
\end{pmatrix}.$ (5)

Note that $\mathcal{T}_{F}$ is neither the time-reversal nor a symmetry
operator of our system. $\mathcal{A}$ is obtained by taking the operator
$\mathcal{T}_{F}$ followed by a $C_{3}$-rotation. Although the operator
$\mathcal{T}_{F}$ works in the four dimensional
$\\{\Ket{\mathbf{n}}\\}$-eigenspace and $C_{3}$ in the real physical space
(under $\\{\Ket{\mathbf{n}}\\}$-basis representation,
$C_{3}\dot{=}diag(C_{3},C_{3},C_{3},C_{3})$), the operator
$\mathcal{A}=C_{3}U_{F}K$ is still antiunitary since $C_{3}U_{F}$ is unitary.

We can now investigate the effect of the application of the operator
$\mathcal{A}$ on a wavefunction $\mathcal{A}\Ket{\psi}=C_{3}U_{F}K\Ket{\psi}$.
The process is graphically explained in Fig. 9. Starting from the four
eigenstates in the first column, the application of $K$ maps them to their
time-reversed counterpart, i.e., from eigenstates with
$\angle\mathbf{k}=\pi/6\mbox{ to }7\pi/6$. Next, the operator $U_{F}$ adds a
negative sign to eigenstates $\Ket{1}$ and $\Ket{3}$, and swaps the pairs
$\Ket{1}\leftrightarrow\Ket{2}$ and $\Ket{3}\leftrightarrow\Ket{4}$, as
visible by comparing the second and third columns of Fig. 9. Afterwards, the
$C_{3}$ operator rotates clockwise each eigenstates about the lattice’s
$C_{3}$ axis by $2\pi/3$, and yields the fourth column in Fig. 9. It can be
shown, by direct comparison with the numerical eigenstates, that this latter
column contains exactly the eigenstates for small $\|\mathbf{k}\|$ along
$\angle\mathbf{k}=\pi$. Therefore, $\mathcal{A}$ maps the eigenstates
$\Ket{\mathbf{n}_{\angle\mathbf{k}=\pi/6}}$ to
$\Ket{\mathbf{n}_{\angle\mathbf{k}=\pi}}$. From now on, for the sake of
brevity, “$\angle\mathbf{k}=$” will be omitted, so that we can write more
compactly,

$\begin{split}\mathcal{A}\Ket{\mathbf{n}_{\pi/6}}&=\Ket{\mathbf{n}_{\pi}},\\\
\mathcal{A}^{-1}\Ket{\mathbf{n}_{\pi}}&=\Ket{\mathbf{n}_{\pi/6}}.\end{split}$
(6)

Multiplying $\mathcal{A}$ from the left to both sides of the eigenvalue
equation
$\mathcal{H}_{\pi/6}\Ket{\mathbf{n}_{\pi/6}}=\omega_{n}\Ket{\mathbf{n}_{\pi/6}}$
yields

$\mathcal{AH}_{\pi/6}\Ket{\mathbf{n}_{\pi/6}}=\omega_{n}\mathcal{A}\Ket{\mathbf{n}_{\pi/6}}=\omega_{n}\Ket{\mathbf{n}_{\pi}}.$
(7)

Also,

$\mathcal{AH}_{\pi/6}\Ket{\mathbf{n}_{\pi/6}}=\mathcal{AH}_{\pi/6}\mathcal{A}^{-1}\Ket{\mathbf{n}_{\pi}}.$
(8)

From the right-hand sides of both equations it is seen
$\mathcal{AH}_{\pi/6}\mathcal{A}^{-1}=\mathcal{H}_{\pi}$, given that we could
also write
$\mathcal{H}_{\pi}\Ket{\mathbf{n}_{\pi}}=\omega_{n}\Ket{\mathbf{n}_{\pi}}$. We
conclude that $\mathcal{A}$ maps the Hamiltonian between different azimuth
angles in $\mathbf{k}$-space for small $\|\mathbf{k}\|$, therefore the
$\Gamma$ point. Hence, ($\|\mathbf{k}\|=0$) is a self-dual point [56] where
$\mathcal{A}$ maps the Hamiltonian back to itself. In other words,
$\mathcal{A}$ is a symmetry operator for the Hamiltonian at the $\Gamma$
point, $\mathcal{AH}_{\Gamma}\mathcal{A}^{-1}=\mathcal{H}_{\Gamma}$, or
$\mathcal{AH}_{\Gamma}=\mathcal{H}_{\Gamma}\mathcal{A}$.

At the $\Gamma$ point, let
$\mathcal{H}_{\Gamma}\Ket{\mathbf{n}_{\Gamma}}=\omega_{n}\Ket{\mathbf{n}_{\Gamma}}$,
then
$\mathcal{H}_{\Gamma}\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}=\mathcal{AH}_{\Gamma}\Ket{\mathbf{n}_{\Gamma}}=\omega_{n}\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}$.
That is, $\Ket{\mathbf{n}_{\Gamma}}$ and
$\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}$ are both eigenstates of
$\mathcal{H}_{\Gamma}$ with the same eigenfrequency $\omega_{n}$. Next, we
need to show that $\Ket{\mathbf{n}_{\Gamma}}$ and
$\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}$ are distinct eigenstates, so that there
is indeed a degeneracy at $\Gamma$. To show that, we need the condition
$\mathcal{A}^{2}\Ket{\mathbf{n}_{\pi/6}}=-C_{3}^{2}\Ket{\mathbf{n}_{\pi/6}}$,
which can be easily verified by applying $\mathcal{A}$ once more, as
demonstrated in Fig. 9.

Before we can show that $\Ket{\mathbf{n}_{\Gamma}}$ and
$\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}$ are distinct, we need to show that for
any two-dimensional complex valued function (such as a wavefunction)
$\Ket{\psi}$, $-C_{3}^{2}\Ket{\psi}\neq\Ket{\psi}$. We can prove it by
negation by first assuming
$\exists\Ket{\psi}:-C_{3}^{2}\Ket{\psi}=\Ket{\psi}$. For brevity, we replace
the notation $C_{3}^{2}$ ($4\pi/3$ clockwise rotation) by $C_{3}^{-1}$
($2\pi/3$ counterclockwise rotation), which are equivalent operations
resulting in identical states. It then follows that

$\displaystyle C_{3}^{-1}\Ket{\psi}$ $\displaystyle=-\Ket{\psi},$ (9)
$\displaystyle C_{3}^{-2}\Ket{\psi}$
$\displaystyle=C_{3}^{-1}(-\Ket{\psi})=+\Ket{\psi},$ (10) $\displaystyle
C_{3}^{-3}\Ket{\psi}$ $\displaystyle=C_{3}^{-1}(+\Ket{\psi})=-\Ket{\psi}.$
(11)

The result in Eq. 11 is in contradiction to the fact that for all
$\Ket{\psi}$, $C_{3}^{-3}\Ket{\psi}=\mathbf{1}\Ket{\psi}$, so the assumption
cannot hold and $\mathcal{A}^{2}\Ket{\psi}=-C_{3}^{2}\Ket{\psi}\neq\Ket{\psi}$
for all $\Ket{\psi}$.

Finally, we can prove that $\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}$ and
$\Ket{\mathbf{n}_{\Gamma}}$ are indeed distinct eigenstates. Using once again
a negation approach, let’s assume that they are the same up to a phase, that
is $\mathcal{A}\Ket{\psi}=e^{+i\gamma}\Ket{\psi}$. Due to the antilinearity of
the operator $\mathcal{A}$,

$\displaystyle\mathcal{A}^{2}\Ket{\psi}$
$\displaystyle=\mathcal{A}(e^{+i\gamma}\Ket{\psi})$ (12)
$\displaystyle=e^{-i\gamma}\mathcal{A}\Ket{\psi}$ (13)
$\displaystyle=e^{-i\gamma}e^{+i\gamma}\Ket{\psi}$ (14)
$\displaystyle=\Ket{\psi},$ (15)

which leads to a direct contradiction to the previous lemma
$\mathcal{A}^{2}\Ket{\psi}\neq\Ket{\psi}$ for all $\Ket{\psi}$. Hence,
$\mathcal{A}\Ket{\mathbf{n}_{\Gamma}}$ and $\Ket{\mathbf{n}_{\Gamma}}$ must be
distinct eigenstates at the same eigenfrequency, hence they are degenerate.
Note that the operator $\mathcal{A}=C_{3}\mathcal{T}_{F}$ is a symmetry
operator only at the $\Gamma$ point, therefore the degeneracy only show at an
isolated point $\Gamma$ rather than indicating globally degenerate eigenstates
as considered in previous studies. In summary, the above discussion follows a
process analogous to that leading to Kramers degeneracies in
spin-$\frac{1}{2}$ systems, but it replaces the time-reversal operator with a
space-time symmetry operator (or, equivalenty, a space-symmetry-adapted time
operator). It is to highlight this conceptual analogy that the term ”synthetic
Kramers pair” was used.

The difference between the symmetry operators protecting the two-fold
degeneracy in either the current elastic lattice
($(C_{3}\mathcal{T}_{F})^{2}=-C_{3}^{2}$ near $\Gamma$) or the electronic
system ($T^{2}=-\mathbf{1}$ globally) also affects the evaluation of the
topological index. This aspect is further discussed in Sec. VI.

A few final remarks are important. When two pieces of materials join together,
depends on whether the two materials are the same or not, an interface or a
dislocation forms between them. In both cases, $C_{3v}$ symmetry is broken
(both globally and locally for the cells by the interface/dislocation). As a
result, the modified version of Kramers degeneracy is lifted due to the
symmetry breaking. This is why in similar photonic and phononic zone-folding
systems, the edge states seem to be always gapped [21, 27, 28]. Nevertheless,
with the $C_{3v}$ symmetry broken, we still found a method to recover the
degeneracy for the counter-propagating edge states, by tuning the local groove
depths $(\delta_{D1},\delta_{D2})$ at the interface (discussed in Sec. XI). We
note that the attained degeneracy at the interface is not a direct result of
the same symmetry operator, since the interface possesses only one mirror
symmetry. Rather, we hypothesize that there must exist a symmetry operator,
defined in the space of the channel depth parameters
$(\delta_{D1},\delta_{D2})$ (see Sec. XI), that maps systems described by
different geometric parameters; this concept would be analogous to results
recently reported by Fruchart et al. [56]. For a certain parameter
configuration, the system could reach the self-dual state, so that the
pseudospin-polarized edge states are decoupled extraordinarily and form
degeneracy at zero $k$.

##  IV Pseudospins

Unlike electrons having intrinsic spin states, the spin of the phonons can
only be well-defined in isotropic media where the longitudinal phonon has spin
0 and the transverse one carries spin 1. In any crystal lattice, provided that
the rotation symmetry is no longer a continuous group, purely longitudinal and
transverse waves do not exist other than along prescribed directions. It
follows that in a lattice, the phonon spin cannot be defined in a rigorous
way, like the electron spin having discrete spin angular momentum [65].

As mentioned in the previous section, once the Kekulé distortion is
introduced, the originally degenerate bands along $\Gamma$-K and
$\Gamma$-${\rm K}^{\prime}$ directions are lifted. The resulting inversion
asymmetry acts as a pseudomagnetic field and splits the pseudospin-pair bands,
$\omega(\mathbf{k,\uparrow})\neq\omega(\mathbf{k,\downarrow})$. Direct
observation of the eigenstates obtained from the numerical results confirms
that the eigenstates of the two split bands have counter-rotating mechanical
energy flux similar to the cyclotron motion of charged particles in a
perpendicular magnetic field. We define the pseudospins directly using the
sign of the pseudocyclotron frequency $\omega_{z}$ based on the mechanical
energy flux circulation. Note that this definition based on direct numerical
calculation can be applied to any eigenstate in the $\mathbf{k}$-space, and it
is not restricted to doubly degenerate $\Gamma$-K or $\Gamma$-${\rm
K}^{\prime}$ directions. The normalized pseudocyclotron frequency is defined
as

$\hat{\omega}_{z}=\frac{\omega_{z}}{\omega}=\frac{1}{\omega}\frac{L_{z}}{I_{zz}},$
(16)

where $\omega_{z}$ is the pseudocyclotron frequency given by $L_{z}/I_{zz}$,
and $L_{z}$ and $I_{zz}$ are the $z$-component of the pseudoangular momentum
and the pseudomoment of inertia, respectively. These quantities are defined as

$\displaystyle L_{z}$
$\displaystyle=\int_{\mathrm{cell}}d^{3}\mathbf{r}~{}(\mathbf{r}\times\mathbf{J})\cdot\hat{\mathbf{z}},$
(17) $\displaystyle I_{zz}$
$\displaystyle=\int_{\mathrm{cell}}d^{3}\mathbf{r}~{}(\|\mathbf{r}\|^{2}-z^{2})w.$
(18)

In the above expressions, $\mathbf{r}=(x,y,z)$ is the position vector with the
origin of the coordinates located at the unit cell center,
$\mathbf{J}=\frac{1}{2}\mathrm{Re}(\bm{\sigma}\cdot\mathbf{v})$ is the period-
averaged mechanical energy flux, while $\bm{\sigma}$ and $\mathbf{v}$ are the
stress tensor and the velocity vector, respectively. In Eq. 18, $w$ is the
period-averaged strain energy density reading
$w=\frac{1}{4}\mathrm{Re}(\bm{\sigma}\cdot\bm{\epsilon})$, and $\bm{\epsilon}$
is the strain tensor. Since the elastic wave is confined in a thin plate, we
only consider the energy flux parallel to the $xy$-plane and the associated
pseudoangular momentum with respect to the out-of-plane $z$-axis. Thus, the
phonon pseudoangular momentum can only have the $z$-component; this is in
contrast with the case of electrons that exhibit an internal spin angular
momentum that can potentially point in any direction, even in 2D materials. We
should also note that the definition we employ for the pseudoangular momentum
is closer to the concept of orbital angular momentum rather than the internal
spin angular momentum in the electronic context.

Note that the pseudocyclotron frequency is introduced in order to facilitate
the nondimensionalization process and it returns an average spinning
frequency, meaning that it accounts for the energy flux of all
pseudocyclotrons included in the integral domain. For simplicity, a Wigner-
Seitz primitive cell centered at a $C_{3}$ symmetry axis (as shown in the
inset Fig. 2 (b) in the main text) is chosen as the reference domain for
integration. The $\omega$-$\mathbf{k}$ dispersion surfaces near the $\Gamma$
point (0.08 $\Gamma$-K) are plotted and color coded using $\hat{\omega}_{z}$.
The results for three configurations identified by the parameters $\delta=0.5$
and $\phi=[-2\pi/3,0,2\pi/3]$ are reported in Fig. 10 (a, b, c), respectively.

Figure 10:  (a-c) The $\omega$-$\mathbf{k}$ dispersion surfaces near the
$\Gamma$ point (0.08 $\Gamma$-K) for $\delta=0.5$ and $\phi=-2\pi/3,0,2\pi/3$,
respectively. The color marks the pseudocyclotron frequency (with the sign
following right-hand rule). (d) Distribution of the strain energy density $w$
along with the mechanical energy flux $\mathbf{J}$ (yellow arrows in
logarithmic scale), on the mid-plane of the phononic plate for a state on the
second lowest band at $0.026~{}\Gamma$-K. The three hexagons with cyan dashed,
white solid, and green dotted borders mark the Wigner-Seitz cells associated
with (a-c), respectively. (e) Schematic showing the r-dependence of the
pseudomagnetic field. The cyan, green, and gray hexagons indicate the three
possible choices of the Wigner-Seitz cells centered at a $C_{3}$ axis. The
blue and red circulating arrows illustrate the pseudocyclotron motion of the
eigenstate in the first and second lowest bands, respectively, assuming a k
point along $\Gamma$-K. The black in- and out-of-plane arrows show the
r-dependence of the pseudomagnetic field. (f) The Berry curvature $\Omega$ of
the second lowest band in log scale. (g) The azimuth-angle-dependence of the
pseudomagnetic field in the k-space near the $\Gamma$ point.

These plots can be leveraged to extract some useful information. First, we
observe that in the presence of Kekulé distortion the frequency bands of the
upper (lower) two neighboring bands are completely detached from each other
except for an isolated degeneracy at the $\Gamma$ point. The repulsion between
the two upper (lower) bands, originally subject to trigonal warping, makes
each of the four new bands acquire a hexagonal warping. Second, in Fig. 10 (a,
c) it is found that the $\hat{\omega}_{z}$ pattern for each band shows a six-
lobe alternating distribution, in which neighboring frequency bands exhibit
opposite $\hat{\omega}_{z}$ values. The pseudoangular momentum has extreme
values along the $\Gamma\textrm{-K}$ and $\Gamma\textrm{-K}^{\prime}$
directions, where the spectrum was doubly degenerate before introducing the
Kekulé distortion, and identically zero along the $\Gamma\textrm{-M}$
directions, where the spectrum was split. From the pseudocyclotron frequency
pattern, we define the pseudospins in the following way: pseudospin up
($\uparrow$) for parts of the frequency band with positive $\hat{\omega}_{z}$
values, and pseudospin down ($\downarrow$) for those with negative
$\hat{\omega}_{z}$ values. Under such definition, a continuous band is not of
a fixed pseudospin but has a six-lobe alternating pseudospin texture. On the
other hand, a pseudospin “band” can be considered to be composed of parts from
two neighboring independently continuous bands. The above discussion can be
visualized in Fig. 11 where, in the frequency bands, red and blue colors
indicate opposite pseudospin textures, and the yellow parts illustrate the
band repulsion due to pseudospin coupling.

Figure 11:  Conceptual illustration showing the formation of the folded band
structure and of the pseudospin coupling in a Kekulé distorted lattice. First,
the double cone composed of opposite pseudospin bands (in red and blue colors)
are originated from the two distinct “original valleys.” Initially, they are
trigonally warped (although the dispersion is circular near the valley, the
lattice symmetry only guarantees $C_{3}$ and $\mathcal{T}$ symmetry, therefore
the frequency band has $C_{6}$ symmetry about $\Gamma$ in the
$\mathbf{k}$-space because $\mathcal{T}$ maps $\mathbf{k}$ to $-\mathbf{k}$).
Second, the folded bands would try to intersect along six different directions
but the coupling between the pseudospins repels the neighboring bands
preventing any intersection, as indicated by the regions in yellow color. As a
result, the two bands are separated except for a degeneracy at the $\Gamma$
point, which is protected by the modified Kramers theorem with the condition
$(C_{3}\mathcal{T}_{F})^{2}=-C_{3}^{2}$ as discussed in Sec. III. Ultimately,
each of the two bands has a six-lobe alternating pseudospin texture, as found
in the numerical results shown in Fig. 10 (a).

From the above observations, it appears that the BHZ Hamiltonian based on two
independent $2\times 2$ massive Dirac Hamiltonians (one per each pseudospin)
cannot describe completely our system and will not allow capturing possible
geometric phases arising from the interaction between the pseudospin bands.

Another interesting observation concerns the cases $\phi=-2\pi/3,0,2\pi/3$.
While, as previously mentioned, they represent the same bulk lattice up to a
rigid-body translation, the corresponding pseudoangular momenta are found to
be different. Indeed, they have the same magnitude but opposite signs for
$\phi=\pm 2\pi/3$, and identically zero values for $\phi=0$. To better
illustrate such peculiarity, we plot the strain energy density $w$ along with
the mechanical energy flux $\mathbf{J}$ (represented by yellow arrows) on the
mid-plane of the phononic plate, as shown in Fig. 10 (d). The plot is taken at
$(k_{x},k_{y})=(10,0)~{}\mathrm{m}^{-1}$ (along $0.026~{}\Gamma$-K) for the
second lowest phononic band. The plot shows that in addition to an average
leftward stream (associated with the negative $x$-gradient of the second
lowest band at $0.026~{}\Gamma$-K), locally circulating patterns distribute
repeatedly throughout the lattice. The three hexagons with cyan dashed, white
solid, and green dotted borders mark the Wigner-Seitz cells (i.e., the domain
of integration) for $\phi=-2\pi/3,0,2\pi/3$, respectively. The inset in Fig.
10 (d) shows a zoom-in near the center of the green hexagon to better
visualize the direction of the flux. Each of the three hexagons covers a full
period of the pattern, but either opposite or net zero circulations about the
centers are found. Since the reference frames and the location of each
pseudocyclotron have no relative motion (i.e. they are fixed in physical
space), the angular momentum of each pseudocyclotron should not vary with
different reference frames. One may wonder if these results violate the
parallel axis theorem of angular momentum? They do not. The reason is that the
pseudocyclotron is periodically distributed, and for each hexagonal domain the
integral takes into account portions from different pseudocyclotrons. Then,
one may ask why not including all the pseudocyclotron contributions in the
same integral domain? It turns out that the improper integral obtained by
extending the integral domain in Eq. 17 to infinity does not converge; this is
conceptually analogous to the divergent alternating series
$\sum_{n=1}^{\infty}(-1)^{n}$ that oscillates endlessly. The previous
discussion shows the intrinsic ambiguity in the pseudoangular momentum so
that, for each band with a certain lattice momentum $(k_{x},k_{y})$, the
pseudospin state is only determinate up to a specific choice of the reference
frame.

In summary, in addition to the discrepancy in the band structure compared with
electronic QSH systems, the pseudospin indices are dependent on the observer
and therefore ambiguous in their definition. In the result section, we will
show that the pseudospin-dependent Chern numbers are in fact gauge dependent.
This latter statement might trigger an additional uncertainty and ambiguity in
the possibility to define edge states. We will show that it is indeed because
of this degree of indeterminacy, due to the dependence on the reference-frame,
that we can conceive edge states along a dislocation boundary between two
domains of the same bulk lattice.

##  V Berry curvature and pseudomagnetic field

In the presence of the two-fold degeneracy at the $\Gamma$ point for opposite
pseudospin bands, a non-Abelian Berry phase is considered and the pseudospin-
dependent Chern numbers form a $2\times 2$ Chern number matrix (CNM)
$C^{\alpha,\beta}=\frac{1}{2\pi}\int d^{2}\mathbf{k}~{}\Omega^{\alpha\beta}$,
where $\alpha,\beta=\uparrow,\downarrow$ indicate the pseudospin index [66].
In the above, the overall integral of the inter-band Berry curvature
$\Omega^{\alpha\beta},\alpha\neq\beta$ yields zero, resulting in a diagonal
CNM (see Sec. VIII for details). Therefore, in the following we will only
focus on the diagonal elements of the Berry curvature matrix. Fig. 10 (f)
shows the Berry curvature $\Omega^{(2)}$, where the superscript indicates the
spectrum index (i.e. the Berry curvature of the second lowest band) in
logarithmic scale. The Berry curvature is strongly localized around the
$\Gamma$ point where it becomes singular. Interestingly, it also exhibits a
six-lobe alternating pattern with extreme values along $\Gamma{\text{-}}{\rm
K}$ and $\Gamma{\text{-}}{\rm K}^{\prime}$ and vanishing values along
$\Gamma{\text{-}}{\rm M}$ directions; this pattern is similar to the one
observed for the pseudocyclotron motion. However, differently from the
pseudocyclotrons, the Berry curvature is a geometric property of the frequency
band associated with the eigenstate evolution in $\mathbf{k}$-space, which is
gauge invariant and does not vary with different reference frames. Such six-
lobe pattern matches that obtained for the pseudospins; equivalently,
$\Omega^{(2)}$ can be considered composed of $\Omega^{\uparrow\uparrow}$ and
$\Omega^{\downarrow\downarrow}$, for each half of the BZ, as the blue and red
parts shown in Fig. 10 (f), respectively. Similar considerations can be drawn
for $\Omega^{(1)}$ by considering inverted pseudospin patterns. The Berry
curvature is known as the $\mathbf{k}$-space magnetic field [67]. We find that
this concept correlates well with the $\mathbf{k}$-space dependence of the
pseudomagnetic field in the real r-space that acts on the cyclotrons and
splits the pseudospin frequency spectra. Figures 10 (e) and (g) show a
schematic illustration of both the r\- and the k-space dependence of the
pseudomagnetic field $\mathbf{B}(\mathbf{k},\mathbf{r})$.

In Fig. 10 (e), the the three cyan, green, and gray hexagons again indicate
the three possible choices (identical to those in Fig. 10 (d)) of the Wigner-
Seitz cells centered at a $C_{3}$ axis. The red and blue circulating arrows
illustrate the pseudocyclotron motion of the eigenstate in the first and
second lowest bands (assuming a k point along $\Gamma$-K), respectively. Given
the known pseudocyclotron distribution, we can easily infer the inhomogeneous
pseudomagnetic field $\mathbf{B}$, as depicted in black by in-plane and out-
of-plane arrows. The pseudomagnetic field must be parallel to the
pseudomagnetic moment $\bm{\mu}$ of the pseudocyclotrons of the first band and
antiparallel to that of the second band since the shifting of the frequency
spectra is $-\bm{\mu}\cdot\mathbf{B}$ (assuming the pseudocyclotron carries
positive “charge,” $\bm{\mu}$ can be inferred by the right-hand rule). We also
note that, for each pseudospin, the total flux through the unit cell is zero,
which is analogous to the quantum Hall effect without Landau levels [9].

Similar to the electron systems, in the phononic pseudospin system the time-
reversal partner of a certain eigenstate $\Ket{n(\mathbf{k},\uparrow)}$, is
$\Ket{n(\mathbf{-k},\downarrow)}$ (except for the $\Gamma\textrm{-M}$
directions where the spin vanishes). For example, the time-reversal partner of
an eigenstate belonging to the second lowest band at some $\mathbf{k}$ point
along $\Gamma{\text{-}}{\rm K}$, is the eigenstate at the $-\mathbf{k}$ point
(along $\Gamma{\text{-}}{\rm K}^{\prime}$) with opposite pseudospin. From the
pseudospin pattern (Fig. 10 (a)), we know it is still on the second lowest
band at the same frequency, which is also a direct result of TRS. It follows
that the pseudospins flip, from momentum $\mathbf{k}$ to $-\mathbf{k}$, and
also for any $\pi/3$ change in the azimuth angle. Now, we can back up the
$\mathbf{k}$-dependence of the pseudomagnetic field that must switch signs for
any $\pi/3$ change in the azimuth angle to yield the opposite spectra
splitting for pseudospin-pair states, as shown in Fig. 10 (g). From the
combination of Fig. 10 (e) and (g), we found the $\mathbf{r}$\- and
$\mathbf{k}$-dependence of the pseudomagnetic field in the current pseudospin
Hall system. In electronic systems, the splitting of two spin bands are
usually described by spin-orbit coupling terms. We note that the distinctive
six-lobe splitting and pseudospin texture showing here are unprecedented from
systems considering conventional Rashba or Dresselhaus spin-orbit coupling
terms[68].

##  VI Local topological order

For electronic QSH systems, the $\mathbb{Z}_{2}$ topological order can be used
to classify the nontrivial topological insulator phase from the ordinary one.
The literature provides several different approaches to the calculation of the
$\mathbb{Z}_{2}$ invariant [55, 66, 69, 70]. Examples include the use of the
wavefunctions of the band pair at the four TR-invariant $\mathbf{k}$ points
(i.e., $\Gamma$ and three M points for triangular lattices) that bound half of
the BZ [55], the integral of either the Berry connection or the Berry
curvature in half of the BZ [69], and the difference in the Chern numbers of
the opposite spin band in a pair [66, 71],
$\frac{1}{2}\left(C^{\uparrow\uparrow}-C^{\downarrow\downarrow}\right)~{}\mbox{mod}~{}2$
for spin conserved systems. The choice of a specific technique depends on
specific symmetries available for the system, as well as on the availability
of the Hamiltonian or the band structure. For the current “zone-folding”
system, given there is no Kramers degeneracy at the M point as we have seen in
Fig. 3 in the main text (recall from Sec. III that in our system,
$(C_{3}\mathcal{T}_{F})^{2}=-C_{3}^{2}$ only holds near $\Gamma$), which does
not guarantee degeneracy at the M point, we cannot make use of the
wavefunctions at those specific $\mathbf{k}$-points to find $\mathbb{Z}_{2}$.
Also, the idea of making use of half of the Brillouin zone, is originated from
$\mathcal{T}^{2}=-\mathbf{1}$ condition, which might not be suitable for our
system. In fact, we can only focus on the vicinity of the $\Gamma$ point, to
define a local topological order that might resemble the $\mathbb{Z}_{2}$
invariant.

Similarly to elastic valley Hall systems [43], in which the resulting valley
dependent Chern numbers are $\pm 1/2$, also in the current system the
pseudospin dependent Chern number is half-quantized
$C^{\uparrow\uparrow/\downarrow\downarrow}=\pm 1/2\textrm{ or}\mp 1/2$, where
$\pm$ or $\mp$ depends on the choice of the unit cell. This is clearly a
direct consequence of the folding which brings the valleys of the initial cell
to coincide at the $\Gamma$ point of the distorted cell. We can therefore
define a local topological order, which conceptually follows the same concept
of a $\mathbb{Z}_{2}$ invariant, in the form
$\nu=\frac{1}{2}(C^{\uparrow\uparrow}-C^{\downarrow\downarrow})$.

Since the two pseudospin dependent Chern numbers have opposite signs ($\pm
1/2$) and switch signs simultaneously under different reference frames, the
evaluated local topological order of a specific lattice can be either
$\nu=+1/2$ or $-1/2$, depending on the choice of reference frames. These
fractional and gauge dependent values indicate that the defined local
topological order is not an exact topological invariant as it is obtained from
the local integral of Berry curvature which does not cover the entire
effective Brillouin zone [70]. In fact, in the “zone-folding” system, we
cannot have any topological phase transition in a strict sense as one can
easily see from Fig. 2 (d) in the main text that a path can be found
connecting any two distinct bulk lattices configurations without crossing the
abscissa $\delta=0$. In other terms, any two bulk lattices can evolve
adiabatically from one to the other without closing the band gap; hence the
exact topological invariant does not change.

However, the local topological order is still a useful indicator that we can
still follow the similar result of the $\mathbb{Z}_{2}$ invariants [1] to
predict the number of Kramers pair edge states on an interface,
$N_{K}=\Delta\nu~{}\mbox{mod}~{}2$, where $\Delta\nu$ is the change in the
local topological order across the interface. For example, similar to the
electronic topological insulator that a pair of edge states can exist on its
outer boundary (i.e., interface between the nontrivial QSH material,
$\mathbb{Z}_{2}=1$, and vacuum, $\mathbb{Z}_{2}=0$), in the “zone-folding”
system, a pair of band-gap-crossing edge states can be synthesised on an
interface connecting two materials with $\nu_{1}=+1/2$ and $\nu_{2}=-1/2$. At
this point, given the sign of local topological order $\nu$ depends on the
choice of the reference frame, it is natural to wonder if an edge state can
exist on the interface between lattices having the same bulk pattern but with
a relative translation (i.e., a dislocation interface). The answer is indeed
affirmative, and a way to conceive a pair of gapless pseudospin-polarized edge
states on a dislocation interface is provided in Sec. XI.

##  VII Pseudospin texture for a general case

We have shown the alternating six-lobe pseudospin texture for the phononic
bands of the lattice with $\phi=0$ ($C_{6v}$). The specific choice of $\phi=0$
was made to create a symmetric dislocation interface and simplify the
experimental setup. Fig. 12 shows the pseudospin texture for the case
$\phi=\pi/6$ ($C_{3v}$) that represents a general case with broken inversion
symmetry. The pseudospin textures of the four bands still exhibits the same
six-lobe pattern, confirming that the symmetric pattern with respect to
$k_{x}$ and $k_{y}$ has a general character rather than representing the
effect of the $C_{6v}$ symmetry of the lattice.

Figure 12:  The pseudospin texture of the lattice with
$(\phi,\delta)=(\pi/6,0.5)$. (a-d) shows the normalized cyclotron frequency
pattern for bands number 1-4, respectively.

##  VIII Numerical method for Berry curvature evaluation

The Bloch eigenstate is obtained from the finite element analysis, in the form
of the displacement field $\mathbf{u}^{(n)}(\mathbf{k},\mathbf{r})$, where the
superscript denotes the spectrum index. We denote
$\Ket{\mathbf{n}(\mathbf{k})}=e^{-i\mathbf{k}\cdot\mathbf{r}}\mathbf{u}^{(n)}(\mathbf{k},\mathbf{r})$
as the “periodic part” of the $n$th Bloch state; similarly
$\Ket{\mathbf{m}(\mathbf{k})}$ for the $m$th state. In the case without
degeneracy of the $n$th band, the eigenstate separated by $\delta\mathbf{k}$
from that at momentum $\mathbf{k}$ can be expressed in terms of the eigenstate
at momentum $\mathbf{k}$ by accounting for an additional phase,
$\Ket{\mathbf{n}(\mathbf{k}+\delta\mathbf{k})}=e^{-i\mathbf{A}^{(n)}\cdot\delta\mathbf{k}}\Ket{\mathbf{n}(\mathbf{k})}$,
where $\mathbf{A}^{(n)}$ is the Abelian Berry connection. With the eigenstate
being normalized, $\Braket{\mathbf{n}}{\mathbf{n}}=1$, we have

$\Braket{\mathbf{n}(\mathbf{k})}{\mathbf{n}(\mathbf{k}+\delta\mathbf{k})}=\Braket{\mathbf{n}(\mathbf{k})}{\mathbf{n}(\mathbf{k})}e^{i\mathbf{A}\cdot\delta\mathbf{k}}=e^{i\mathbf{A}\cdot\delta\mathbf{k}}.$
(19)

For infinitesimal $\delta\mathbf{k}$, both sides of the above equation can be
simplified (by taking a series expansion up to the linear term in
$\delta\mathbf{k}$) as

$\begin{array}[]{cccccc}lhs=&\Braket{\mathbf{n}(\mathbf{k})}{\mathbf{n}(\mathbf{k})}&+&\Braket{\mathbf{n}(\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{n}(\mathbf{k})}\cdot\delta\mathbf{k}&+&O(\delta
k^{2}),\\\ rhs=&1&+&i\mathbf{A}\cdot\delta\mathbf{k}&+&O(\delta
k^{2}).\end{array}$ (20)

Comparing the linear terms in both equations, the Berry connection can be
written in the well-known form
$\mathbf{A}^{(n)}(\mathbf{k})=-i\Braket{\mathbf{n}(\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{n}(\mathbf{k})}$.
In our numerical implementation, where we must use a small but finite-sized
$\delta\mathbf{k}$, we use the form shown in Eq. 19 that yields better
accuracy and numerical robustness when finite difference approximation is
used. The finite (central) difference formula to compute the $k_{x}$ and
$k_{y}$ components of the Berry connection,

$\displaystyle\begin{split}A^{(n)}_{x}(\mathbf{k})&=\frac{\angle\Braket{\mathbf{n}(\mathbf{k}-\delta
k\,\hat{\mathbf{k}}_{x})}{\mathbf{n}(\mathbf{k}+\delta
k\,\hat{\mathbf{k}}_{x})}}{2\delta k},\\\
A^{(n)}_{y}(\mathbf{k})&=\frac{\angle\Braket{\mathbf{n}(\mathbf{k}-\delta
k\,\hat{\mathbf{k}}_{y})}{\mathbf{n}(\mathbf{k}+\delta
k\,\hat{\mathbf{k}}_{y})}}{2\delta k},\end{split}$ (21)

where “$\angle$” extracts the argument of a complex number (mod $2\pi$),
$\delta k$ is the step size, and $\hat{\mathbf{k}}_{x},\hat{\mathbf{k}}_{y}$
are the unit vectors in the $\mathbf{k}$-space. The bra-ket inner product sums
up the integrals of the three components of the displacement field of the
corresponding eigenstates over a unit cell in $\mathbf{r}$-space.

We note that the eigenstates all have the $U(1)$ gauge ambiguity, and the
evaluation of Eq. 21 requires a smooth gauge choice. To guarantee the
smoothness, we pre-process the eigenstates returned from COMSOL Multiphysics
with an additional $U(1)$ gauge transformation,
$\Ket{\mathbf{n}(\mathbf{k})}\rightarrow
e^{i\theta^{(n)}(\mathbf{k})}\Ket{\mathbf{n}(\mathbf{k})}$, where
$\theta^{(n)}(\mathbf{k})=\angle u_{z}^{(n)}(\mathbf{r}_{\rm ref},\mathbf{k})$
is a reference phase which is taken from the argument of the $z$-component of
the displacement at a fixed reference point in the unit cell. Given that, for
every smooth gauge choice, $u_{z}^{(n)}(\mathbf{r}_{\rm ref},\mathbf{k})$ must
be smooth throughout the $\mathbf{k}$-space, fixing its argument to zero
determines a smooth gauge choice. Note that any reference point could be
selected, as long as $u_{z}^{(n)}(\mathbf{r}_{\rm ref},\mathbf{k})$ does not
vanish at some $\mathbf{k}$ (hence rendering the Berry connection gauge
dependent). The gauge invariant Berry curvature is then obtained as the curl
of the Berry connection, also using the finite difference formula,

$\displaystyle\begin{split}\Omega^{(n)}&=\frac{A_{y}^{(n)}(\mathbf{k}+\delta
k\,\hat{\mathbf{k}}_{x})-A_{y}^{(n)}(\mathbf{k}-\delta
k\,\hat{\mathbf{k}}_{x})}{2\delta k}\\\ &-\frac{A_{x}^{(n)}(\mathbf{k}+\delta
k\,\hat{\mathbf{k}}_{y})-A_{x}^{(n)}(\mathbf{k}-\delta
k\,\hat{\mathbf{k}}_{y})}{2\delta k}\end{split}$ (22)

Considered the degeneracy at the $\Gamma$ point,
$\omega^{(1)}=\omega^{(2)},\omega^{(3)}=\omega^{(4)}$, the non-Abelian Berry
connection should be considered. The latter definition takes into account the
interaction between two neighboring bands near the $\Gamma$ point and can be
obtained by naturally extending Eq. 21 to form a $2\times 2$ matrix [66],

$\displaystyle\begin{split}A^{(m,n)}_{x}(\mathbf{k})&=\frac{\angle\Braket{\mathbf{m}(\mathbf{k}-\delta
k\,\hat{\mathbf{k}}_{x})}{\mathbf{n}(\mathbf{k}+\delta
k\,\hat{\mathbf{k}}_{x})}}{2\delta k},\\\
A^{(m,n)}_{y}(\mathbf{k})&=\frac{\angle\Braket{\mathbf{m}(\mathbf{k}-\delta
k\,\hat{\mathbf{k}}_{y})}{\mathbf{n}(\mathbf{k}+\delta
k\,\hat{\mathbf{k}}_{y})}}{2\delta k},\end{split}$ (23)

The Berry curvature matrix can be obtained accordingly.

To better resolve the Berry curvature close to the singular $\Gamma$ point, we
evaluate the Berry curvature in a step-wise fashion using different step
sizes: as getting closer to the $\Gamma$ point, we evaluate Berry curvature
using smaller step size in $\mathbf{k}$-space, which it refines by a factor of
two for every next level. The refinement is applied for eight levels (enough
to guarantee the convergence of the numerical integral) as shown in Fig. 13.

Figure 13:  The mesh grids used to evaluate the Berry curvature in
$\mathbf{k}$-space. Eigenstates, Berry connection and curvature are evaluated
at the grid points.

##  IX Chern number matrix

The element of the $2\times 2$ Chern number matrix (CNM) is
$C^{\alpha,\beta}=\frac{1}{2\pi}\int d^{2}\mathbf{k}\ \Omega^{\alpha,\beta}$,
where $\alpha$ and $\beta$ denote the pseudospin rather than the spectrum
indices. For example, considering the lower cone (spectrum indices $=1,2$),
along $+\Gamma K$, $\Omega^{\uparrow\downarrow}=\Omega^{(1,2)}$. Note that, in
an equivalent fashion, we can write
$\Omega^{\uparrow\downarrow}=\Omega^{(2,1)}$ along $-\Gamma K$.

The total spin-related Chern number can be explicitly written as
$C_{t}=C^{\uparrow\uparrow}+C^{\uparrow\downarrow}-C^{\downarrow\uparrow}-C^{\downarrow\downarrow}$.
We claim, and will show later, that although in our system the pseudospins are
not conserved, the off-diagonal elements of the CNM (i.e.,
$C^{\uparrow\downarrow},C^{\downarrow\uparrow}$) vanish, therefore we can take
the local topological order
$\nu=\frac{1}{2}\left(C^{\uparrow\uparrow}-C^{\downarrow\downarrow}\right)$.

Let us consider an off-diagonal element of the Berry connection,
$\mathbf{A}^{\uparrow\downarrow}(\mathbf{k})=-i\Braket{\uparrow(\mathbf{k})}{\nabla_{\mathbf{k}}}{\downarrow(\mathbf{k})}$.
At some $\mathbf{k}$ point, it is also
$\mathbf{A}^{\uparrow\downarrow}(\mathbf{k})=-i\Braket{\mathbf{m}(\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{n}(\mathbf{k})}$.
At $-\mathbf{k}$, since the pseudospin indices swapped due to the odd texture,
$\mathbf{A}^{\uparrow\downarrow}(-\mathbf{k})=-i\Braket{\mathbf{n}(\mathbf{-k})}{\nabla_{\mathbf{k}}}{\mathbf{m}(-\mathbf{k})}$.
Observing that $\Ket{\mathbf{n}(-\mathbf{k})}$ is the time-reversed version of
$\Ket{\mathbf{n}(\mathbf{k})}$ and we can take
$\Ket{\mathbf{n}(-\mathbf{k})}=\mathcal{T}\Ket{\mathbf{n}(\mathbf{k})}=\Ket{\mathbf{n}^{\ast}(\mathbf{k})}$
where ∗ means the complex conjugation of a wavefunction or of a scalar, we
have

$\displaystyle\mathbf{A}^{\uparrow\downarrow}(-\mathbf{k})$
$\displaystyle=-i\Braket{\mathbf{n}(-\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{m}(-\mathbf{k})}$
$\displaystyle=-i\Braket{\mathbf{n}^{\ast}(\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{m}^{\ast}(\mathbf{k})}$
$\displaystyle=-i\Braket{\mathbf{n}(\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{m}(\mathbf{k})}^{\ast}$
$\displaystyle=-i\Braket{\mathbf{m}(\mathbf{k})}{\nabla_{\mathbf{k}}}{\mathbf{n}(\mathbf{k})}=\mathbf{A}^{\uparrow\downarrow}(\mathbf{k})$

The above result means that the off-diagonal elements of the Berry connection
are even functions, therefore their derivatives
($\partial_{k_{p}}A_{q}^{\uparrow\downarrow},\partial_{k_{p}}A_{q}^{\downarrow\uparrow}$
where $p,q=x~{}{\rm or}~{}y$) must be odd functions. It follows that the off-
diagonal elements of the Berry curvature are odd functions, therefore their
integrals around the $\Gamma$ point vanish and the CNM in our system is a
$2\times 2$ diagonal matrix.

##  X Topology of the $(\phi,\delta)$ parametric space

We note that the $(\phi,\delta)$ parametric space shown in Fig. 2 (d) in the
main text can be visualized in a more natural way (already showing evidence of
the six-lobe band properties). The upper and lower bounds of the parameter
$\phi\in[0,\pi)$ can be connected to form a loop while at the same time the
parameter $\delta\in\left[-1,1\right]$ is inverted three times, given that the
same bulk lattice configuration appears every $\pi/3$ increments with opposite
$\delta$. Therefore, the parametric space can be pictured as a 1.5-turn
($3\pi$)-twisted Möbius strip as shown in Fig. 14 (a), where the blue line
indicates $\phi=0$ and the black mid-strip ring is the $\delta=0$ curve.

Figure 14:  A different representations of the $(\delta,\phi)$ parametric
space. (a) The parametric space can be pictured as a 1.5-turn ($3\pi$)-twisted
Möbius strip. The upper and lower bounds of the parameter $\phi\in[0,\pi)$ can
be connected to form a loop while at the same time the parameter
$\delta\in\left[-1,1\right]$ is inverted three times, given that the same bulk
lattice configuration appears every increased $\pi/3$ in with opposite
$\delta$. The blue line indicates $\phi=0$ and the black mid-strip ring is the
$\delta=0$ curve. (b) The strip with the mid-ring shrinking to a point, since
all points toward the $\delta=0$ curve converge to the same reference
configuration. (c) A “top view” of (b).

Given that all points toward the $\delta=0$ curve converge to the same
configuration (that is the undistorted reference lattice), the mid-ring should
shrink to a point, as shown in Fig. 14 (b). A “top view” with the azimuth
angle being twice the parameter $\phi$ is shown in Fig. 14 (c). Note that an
increment of an arbitrary $\Delta\phi$ in $\phi$ does not mean a
$\Delta\phi$-rotated lattice or $\mathbf{k}$ vector. As mentioned in an
earlier paragraph, for any given $\phi$, the new parameter $\phi+2\pi/3$
yields a $2\pi/3$ rotation in the entire lattice, or $-2\pi/3$ rotation in the
$\mathbf{k}$ vector. For example, the center of the three red lobes in Fig. 14
correspond to $\phi=0,2\pi/3,4\pi/3$. Therefore, they can also be interpreted
as the $\phi=0$ configuration with the reference frame ($\mathbf{k}$ vector)
rotated by $0,-2\pi/3,-4\pi/3$. They should all yield the same gauge-invariant
Berry curvature. On the other hand, the opposite blue lobes correspond to the
same value of $\phi$ with an additional negative sign in $\delta$ which
indicates a Berry curvature with opposite sign. At the same time, the
additional negative sign in $\delta$ can also be identically represented by
the $\phi=0$ configuration with $\pi$-rotated $\mathbf{k}$ vector. Therefore,
we have linked the distortion parameter $\phi$ with the azimuth angle in the
$\mathbf{k}$-space, and the topology of the parametric space already suggests
the $\mathbf{k}$-space dependence of the Berry curvature.

##  XI Dislocation interface to achieve decoupled gapless edge states

To create a dislocation that is also a plane of mirror symmetry, we select the
configuration with $\phi=0$ so that the bulk lattice possesses $C_{6v}$
symmetry and its mirror image with respect to any plane normal to the $y$-axis
still possesses the same bulk pattern. The dislocation is then aligned with an
$x$-parallel groove (with distortion strength $\delta_{2}$) to retain the
overall honeycomb-like pattern across the dislocation, as shown in Fig. 15
(a). However, the two counter-propagating edge states are strongly coupled,
exhibit mixed pseudospin polarization, and are gapped.

Figure 15:  The symmetric dislocation interfaces. (a) The dislocation aligned
with an $x$-parallel groove with distortion strength $\delta_{2}$. The edge
states on such dislocation interface are gapped. (b) A modified dislocation
interface. The groove depths at the two nodes along the interface are
optimized as $\delta_{D1}=0.4$ and $\delta_{D2}=-0.31$ so that the counter-
propagating edge states are gapless and fully decoupled.

Hence, we allow the groove nodes along the dislocation to take different
depths, as illustrated in Fig. 15 (b). We sweep through different values of
the distortion strength at the dislocation $(\delta_{D1},\delta_{D2})$ and
find a (non-unique) set of values $(0.4,-0.31)$ such that the edge states’ gap
closes without mixing the pseudospins, as shown in Fig. 4 in the main text.

##  XII Fabrication and experiment setup

An $18^{\prime\prime}\times 18^{\prime\prime}\times 1/4^{\prime\prime}$
Mic6${}^{\mbox{\textregistered}}$ aluminum cast plate was used to fabricate
the phononic plate. The cast aluminum alloy plate provides better flatness and
lower residual stress over cold-rolled aluminum sheets, which ultimately
reduces warpage after machining. The honeycomb groove patterns on both sides
were engraved by a ball-end mill programmed to cut through the defined routes
on a CNC mill. A maximum warping deflection of $0.017^{\prime\prime}$ at the
plate center was found after machining one side of the plate. This resulted in
overcutting in the groove depth on the other side, which ultimately produced a
slight asymmetry with respect to the mid-plane. A 2-inch margin on the
boundary of the plate was left untreated (i.e. not machined) so to apply a 3M™
Damping Foil 2552 to reduce unwanted wave reflection from the edges during the
test. Two piezoelectric ceramic lead zirconate titanate (PZT-5A) plates were
used as actuators. The electric signal was amplified by a
Trek${}^{\mbox{\textregistered}}$ PZD350A M/S amplifier. A Polytec PSV-500
scanning laser-Doppler vibrometer system was used to measure the response of
the phononic plate.

We also conducted a time-transient measurement following an input wave packet
signal centered at 172 kHz as shown in Fig. 16 (a). Instantaneous velocity
profiles at selected time frames are shown in Fig. 16 (b-i). Results evidently
show that the wave packet propagates through the two acute corners without
observable reflections. The faster wave with longer wavelength (visible in
early time frames (b-d)) is the symmetric bulk mode triggered during the ramp
up phase of the transient. A full video of the transient response is also
provided in the online Supporting Information [61].

Figure 16:  The response of the phononic plate to a transient excitation. (a)
The wave packet used as input signal. (b-i) Instantaneous velocity profiles at
selected time frames. The wave packet can travel undisturbed through the two
acute corners without observable reflections.

The results shown in Fig. 5 (c,d) in the main text were obtained using a white
noise input signal, and the frequency response was obtained by measuring the
velocity spectrum normalized by the input voltage spectrum. In the transient
measurement, a Kaiser window was used to generate the wave packet. However,
due to the capacitive load of the PZT plate, the signal was moderately
distorted. The signal shown in Fig. 16 (a) is the monitored output signal of
the amplifier, that is the actual signal applied to the actuator.

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