††thanks: These three authors contributed equally††thanks: These three authors
contributed equally††thanks: These three authors contributed equally

# Smart patterning for topological pumping of elastic surface waves

Shaoyun Wang  Department of Mechanical and Aerospace Engineering, University
of Missouri, Columbia, MO 65211, USA  Zhou Hu  School of Aerospace
Engineering, Beijing Institute of Technology, Beijing 100081, China  Qian Wu
Department of Mechanical and Aerospace Engineering, University of Missouri,
Columbia, MO 65211, USA  Hui Chen  Piezoelectric Device Laboratory, School of
Mechanical Engineering and Mechanics,
Ningbo University, Ningbo 315211, China  Emil Prodan  Department of Physics,
Yeshiva University, New York, NY 10016, USA  Rui Zhu<EMAIL_ADDRESS>School
of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081,
China  Guoliang Huang<EMAIL_ADDRESS>Department of Mechanical and
Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

###### Abstract

Topological pumping supplies a robust mechanism to steer waves across a sample
without being affected by disorders and defects. For the first time, we
demonstrate the pumping of elastic surface waves, achieved by a smart
patterning of a surface that creates a synthetic dimension, which is explored
by the wave as it is launched perpendicularly to the steering direction.
Specifically, we design and fabricate an elastic medium decorated with arrays
of pillar-type resonators whose eigenmodes are locate below the sound cone,
together with coupling bridges edged according to a specific algorithm. We
establish a connection between the collective dynamics of the pillars and that
of electrons in a magnetic field by deriving an accurate tight-binding model
and developing a WKB-type analysis suitable for such discrete aperiodic
systems with spatially slow-varying couplings. This enable us to predict
topological pumping pattern, which is numerically and experimentally
demonstrated by steering waves from one edge of the system to the other.
Finally, the immune character of the topologically pumped surface waves
against disorder and defects is evidenced. The principle of surface patterning
together with the WKB-analysis could provide a powerful new platform for
surface wave control and exploration of topological matter in higher
dimensions.

††preprint: APS/123-QED

##  I Introduction

Topological matter is a rapidly growing field in which topological concepts
are exploited to discover and classify new phases of matter [1, 2, 3, 4]. In
this context, a hallmark achievement was the discovery of the integer quantum
Hall effect [citation]. In the past decade, topological phases analogous to
quantum Hall insulators have been engineered across a wide range of time-
modulated platforms, including electronics [5, 6, 7], photonics [8, 9, 10, 11,
12, 13], acoustics [14, 15, 16, 17], and mechanics [18, 19, 20, 21, 22, 23].
The existence of the conventional gapless edge states and surface states is
guaranteed by the bulk-boundary correspondence. These time-dependent systems
can provide outstanding opportunities not possible with passive materials,
enabled by the high controllability and flexibility of these platforms.
However, a physical realization of a dynamically controlled topological
pumping that produces topological transport is extremely challenging because
external or active physical fields are typically needed [24].

To overcome the challenges associated with time-modulated system, rendering
synthetic dimensions via space modulations was recently suggested because it
does not require any active materials or other external mechanisms to break
the time-reversal symmetry [19, 20]. The phases of the space-modulations can
be used as adiabatic parameters that augment the physical space [22]. It is
intriguing to see these phases as additional global degrees of freedom,
usually called phasons, living on a torus. The central idea of synthetic
dimensions is to exploit and harness such degrees of freedom with atoms,
photons or phonons to mimic the dynamic motion along extra spatial directions.
The key advantage of synthetic dimensions is that pumping parameters can be
engineered very naturally in the strength of the couplings along the extra
dimension. Synthetic dimensions have led to new discoveries of the 2D and 4D
quantum Hall systems in ultracold atomic gases [25, 26, 27], photonics [28,
29, 30, 31] and acoustics and mechanics [32, 33, 34] due to their flexibility.
Rendering of the synthetic space is growing into one of most appealing
approach to control and steer topological wave transport in different systems.

Surface elastic waves are a class of polarized waves that propagate on the
surface of a semi-infinite elastic medium. They are confined within a
superficial region whose thickness is comparable with their wavelength [35].
Manipulating surface waves has been of considerable interest with widespread
applications in earthquake mitigation, nondestructive evaluation, wave
filtering and sensing [36, 37, 38]. Based on the Bragg scattering and local
resonance mechanisms, manipulation and control of surface waves has been
recently investigated in the phononic and metamaterial community for various
applications such as exotic wave transmission and reflection, wave focusing
and cloaking [39, 40, 41]. Among existing approaches, the metamaterial with
pillar-type resonators is regarded as one of the most promising microstructure
designs because of their simple structure and process-friendly fabrication.
However, it is not trivial to apply pillar-type metamaterials for the
topological surface wave transport. Indeed, there is of fundamental and
practical significance to physically realize space-modulated pillar-type
metamaterials for topological surface wave transport along desired orbits [42,
43].

In this study, we present theoretical, numerical and experimental
investigations of Rayleigh wave topological pumping by leveraging a pillar-
based platform with space modulations. The proposed structures can be
described as aperiodic mechanical wave-channels carrying different phason
values that are stacked and coupled with each other. By slowly varying the
phason along the stacking direction, we demonstrate here that, with such an
approach, we can explore any continuous orbit inside the phason space, and
even control the speed along the path to shape the surface pumped pattern. As
a result, we can render these abstract trajectories, occurring in the
synthetic dimensions, on the physical dimension along the stackings. In turn,
this enables us to control the propagation of the surface waves in space as
well as the temporal phases of the signals.

With the control over the phason, we experimentally demonstrate edge-to-edge
topological wave pattern on the space-modulated mechanical metasurface, which
is robust against random fluctuations in the couplings. The analytical study
of pumping process under adiabatic condition is formulated by using the WKB
approximation and the modulation functions of parameters with nontrivial
topological phase is also analytically obtained. Based on that, we further
explore various ways in which we can control these pumping processes and
validate topological mode steerings in time-domain simulation. It is believed
that our work breaks ground for engineering applications, where the couplings
in a space-modulated mechanical metasurface can be programmed for selective
and robust point-to-point transport of surface wave signals.

Figure 1: Design principle and dispersion analysis. a Schematic illustration
of the topological surface wave transport system. Each row in $x$ corresponds
to a supercell that includes three unit cells (inset). b Photograph of the
experimental sample fabricated out of aluminum by a milling machine. c
Numerically obtained dispersion curves (blue dots) for the unit cell with
$\phi_{j}=\pi$. The orange curves represent the dispersion curves of the
discrete mass-spring model obtained by numerical fitting. The gray regions are
filled with bulk modes. Their interfaces with the surface wave region define
the sound cone. d Dispersion diagram for the supercell terminated by free
boundary conditions in the $x$-direction and Floquet boundary conditions in
the $z$-direction. The edge-bulk-edge (EBE) mode is represented by the magenta
surface, whereas the bulk bands are indicated by gray surfaces. The orange cut
plane corresponds to the excitation frequency $f_{c}=41.88\text{Hz}$. The
interaction curve between excitation frequency plane and EBE surface gives the
instantaneous wave number $q(z)$ on which the circle is right edge mode, the
triangle is the bulk mode and the square is the left edge mode. e The top,
middle and bottom panels are the corresponding eigenmodes of supercell at
$\phi=0.6\pi,\,\pi,\,\text{and}\,1.4\pi$ with $q=\pi/a$ in d.

##  II Results

Physical rendering of synthetic spaces. We start by explaining the principles
of physical rendering of synthetic spaces in the context of surface wave
transport. Figure 1a, b show our surface wave platform featuring a planar
array of elastic pillar-type resonators coupled horizontally and vertically
through thin plates (see Methods sample preparation for fabrication details).
Each resonator is assigned an address $(i,j)\in\mathbb{Z}^{2}$ in the $x$-$z$
plane. The heights of the connecting plates in the $x$-direction are modulated
according to the protocol $h_{ij}=h_{0}[1+\Delta_{0}\cos(2\pi i/3+\phi_{j})]$,
while the geometry of the connecting thin plate along $z$ direction is uniform
across the sample. Any such modulation has a phase that can take any value in
the abstract interval $[0,2\pi]$, representing here the synthetic space. In a
time-modulated setting, one will dynamically drive the phase $\phi$ by rapid
re-configurations of the systems [44]. Instead, by setting the phason value of
the $j$-th row as $\phi_{j}(z)=\phi_{s}+(\phi_{f}-\phi_{s})\frac{j}{N}$ with
$N$ being the total number of rows, we effectively render the synthetic space
along the $z$-axis. The parameters will be fixed as $\phi_{s}=0.6\pi$ and
$\phi_{f}=1.4\pi$.

As shown in Fig. 1a, each $x$-directional row displays a unit cell containing
three pillars. The dispersion curves of the unit cell obtained with COMSOL
Multiphysics is shown in Fig. 1c. The computation was carried out by imposing
Floquet boundary conditions in both $x$\- and $z$-directions. Since the
modulation amplitude $\Delta_{0}$ is small, $\phi_{j}$ is irrelevant for the
dispersion curves and can be assumed as 0. Below the sound cone (white
region), one can see three surface wave branches, whose eigenmodes are
localized on the surface and decay quickly into the bulk; see also
Supplementary Information for detailed illustrations of the corresponding mode
shapes. The region above the sound cone, shown in gray, is referred to as the
”bulk modes” region.

Figure 2: Topological surface pumping on the elastic surface with space
modulated pillars. a,b Experimental a and numerical b modal profile of out-of-
plane displacement field at the frequencies $f=$42.45 kHz and $f=$41.88 kHz,
respectively, by piezoelectric patch (gray part in the bottom of the cuboid)
excitation. c Frequency spectrum of the resonator with indices $i=2\
\text{and}\ j=19$ from experimental measurement. The resonance peak noted by
red dot is the EBE mode. d The wavelet transform of the eigenmode from
numerical simulation along synthetic dimension. The purple curve is the
interaction curve from Fig. 1d. e The mode decomposition of the displacement
field in the right panel of a of each chain for different $z$.

To facilitate the physical interpretation of the surface wave pumping and
illuminate the function of the phason, we develop a discrete mass-spring model
for the surface wave eigenmodes using the mode-coupling theory. This model
takes the form of the following difference equation for the amplitudes
$\psi_{i,j}$ of the local resonances carried by the individual pilars (see
Supplementary Information for derivation details)

$\displaystyle\kappa^{0}\psi_{i,j}+\kappa^{h}_{i-1,j}\psi_{i-1,j}+\kappa^{h}_{i,j}\psi_{i+1,j}$
(1)
$\displaystyle\quad+\kappa^{v}[\psi_{i,j-1}-2\psi_{i,j}+\psi_{i,j+1}]=-M\omega^{2}\psi_{i,j}.$

Here, $M$, $\kappa^{0}$, $\kappa^{v}$ and $\kappa^{h}$ are interpreted as the
effective mass and grounded, vertical and horizontal spring stiffnesses of the
model, respectively. The values of these effective parameters are determined
by fitting the dispersion curves of the continuous model (blue dots in Fig.
1c). Specifically, we obtain $M=1~{}\text{kg}$,
$\kappa^{0}=49.6~{}\text{GN/m}$, $\kappa^{v}=1.9~{}\text{GN/m}$, and
$\kappa^{h}_{i,j}=\kappa^{h}_{0}\left[1+\Delta\cos(2i\pi/3+\phi_{j})\right]$,
where the modulation coefficients read $\kappa^{h}_{0}=5.5~{}\text{GN/m}$ and
$\Delta=0.67$. As shown in Fig. 1c, the continuous and discrete dispersion
curves exhibit satisfactory agreement, thereby demonstrating the reliability
of the discrete model.

WKB-type analysis. By replacing the index $j$ with the coordinate $z=ja$, we
rewrite $\psi_{i,j}=\psi_{i}(z)$ and $\phi_{j}=\phi(z)$, as well as

$\kappa^{h}_{i,j}=\kappa^{h}_{i}(z)=h_{0}[1+\Delta_{0}\cos(2\pi
i/3+\phi(z))].$ (2)

We also introduce the second-order central difference operator

$\delta^{2}f(z)=\frac{f\left(z+a\right)-2f(z)+f\left(z-a\right)}{a^{2}}.$ (3)

In addition, a vector
$\bm{\psi}(z)=[\psi_{0}(z),\psi_{1}(z),...,\psi_{3M}(z)]^{\text{T}}$ is
defined including all the mode coefficients. By doing so, the dispersion
equation (1) can be written very compactly as

$a^{2}\kappa^{v}\delta^{2}\bm{\psi}(z)+[\textbf{K}(z)+\omega^{2}]\bm{\psi}(z)=0,$
(4)

in which $\textbf{K}(z)$ is the matrix with the entries

$\text{K}_{ik}(z)=\kappa^{0}\delta_{ik}+\kappa^{h}_{i}(z)\delta_{i,k+1}+\kappa^{h}_{i}(z)\delta_{i+1,k},$
(5)

where $\delta_{ik}$ is the Kronecker delta. Equation (4) is very close in
spirit with the Schroedinger equation appearing in the setting of WKB
approximation theory [45, 20]. The difference is that, instead of dealing with
a potential, we are dealing with the non-diagonal matrix K(z) which,
nevertheless, is slowly varying with $z$. In this regime, the following WKB-
type expansion is justified

$\bm{\psi}(z)=e^{i\theta(z)/a}[\bm{\psi}^{(0)}(z)(z)+a\bm{\psi}^{(1)}(z)(z)+\cdots],$
(6)

and, by keeping track of the powers of $a$, we can derive the exact equations
satisfied by each $\bm{\psi}^{(\alpha)}$. In particular, we find for the
leading term that this equation is (see Supplementary Information)

$\big{(}\bm{K}(z)+\omega^{2}\big{)}\bm{\psi}^{(0)}(z)=4\sin^{2}\left(\frac{\delta\theta(z)}{2}\right)\bm{\psi}^{(0)}(z),$
(7)

where $\delta\theta(z)=[\theta(z+a/2)-\theta(z-a/2)]/a$. This equation has
solutions of the form

$\bm{\psi}_{n}(z)=A_{n}(z)e^{\text{i}\sum_{\xi=0}^{\xi=z}q_{n}(\xi)}\bm{\varphi}_{n}(z)+o(a),$
(8)

where $\bm{\varphi}_{n}(z)$ is the $n$-th eigenmode of the $\textbf{K}(z)$
matrix

$\textbf{K}(z)\bm{\varphi}_{n}(z)=-\mu_{n}(z)\bm{\varphi}_{n}(z),$ (9)

at row $z$ and $q_{n}(z)$ satisfies the equation

$4\kappa^{v}\sin^{2}\frac{q_{n}(z)}{2}+\mu_{n}(z)=\omega^{2}.$ (10)

As in the standard WKB-theory [45], an analysis at the order-one level of the
asymptotic expansion enables us to pinpoint the $z$-dependence amplitude
$A_{n}(z)$ (see Supplementary Information), and to finally present the
complete set of solutions for the dispersion equation (4)

$\bm{\psi}_{n}(z)=\frac{c_{n}}{\sqrt{\omega^{2}-\mu_{n}(z)}}e^{\text{i}\sum_{\xi=0}^{\xi=z}q_{n}(\xi)}\bm{\varphi}_{n}(z)+o(a),$
(11)

We recall that the derivation of these solutions relies only the adiabatic
evolution of the phason with $z$ and no considerations of long-wavelengths or
paraxial approximation were made. Thus, our results cover the short-wavelength
and nonparaxial regions. Lastly, since our samples are finite, we need to
impose free boundary conditions on the top and bottom boundaries in the
$z$-direction. In this case, the mode shape of the $n$th eigenmode is in the
form of

$\bm{\psi}_{n}(z)=\frac{c_{n}\sin Q_{n}(z)+d_{n}\cos
Q_{n}(z)}{\sqrt{\omega^{2}-\mu_{n}(z)}}\bm{\varphi}_{n}(z),$ (12)

where $c_{n}$ and $d_{n}$ are coefficients of superposition and
$Q_{n}(z)=\sum_{\xi=0}^{\xi=z}q_{n}(\xi)$ is the dynamical phase produced by
our derivation.

Figure 3: Time response of the topological surface wave transport. a-c.
Magnitudes of displacement field at $0.5~{}\text{ms}$, $2.5~{}\text{ms}$ and
$4~{}\text{ms}$, respectively. A 50-cycle tone burst excitation centered at
41.88 kHz is applied on the bottom supercell

Topological pumping. The complete set of solutions (11) indicates that, when
the meta-surface is excited at pulsation $\omega$ with a source placed at
position $z=0$, it will resonate very strongly with the mode that has its
resonant frequency $\mu_{n}(z=0)$ close to $\omega^{2}$. Thus, we have a
mechanism to selectively load a specific mode out of a fairly rich set of
resonant modes. Furthermore, Eq. (11) indicates that, with such a source
turned on, upon the inspection of row $z$, we will see the eigenmode
$\mu_{n}(z)$ of the one-dimensional tight-binding operator $\textbf{K}(z)$ (up
to a multiplicative factor). Since $\textbf{K}(z)$ depends only on the phason
value $\phi(z)$, i.e. $\textbf{K}(z)=\textbf{K}_{\phi(z)}$, one can now see
explicitly how the dependence of the spectral properties of
$\textbf{K}_{\phi}$ on the phason has been rendered along the $z$-coordinate,
for us to experience, measure and use its resonant modes in future
applications. Furthermore, by design, the phason is being pumped from
$\phi_{i}$ to $\phi_{f}$ as the structure is examined from bottom ($z=0$) to
the top ($z=Na$).

We now turn our attention to $\textbf{K}_{\phi}$ defined by the equations (2)
and (5), and process it as

$\text{K}_{ik}(\phi)=\kappa^{h}_{i}(\phi)[\delta_{i,k+1}+\delta_{i+1,k}+\kappa^{0}\tilde{\kappa}^{h}_{i}(\phi)\delta_{ik}],$
(13)

where

$\tilde{\kappa}^{h}_{i}=1/\kappa^{h}_{i}\approx 1/h_{0}[1-\Delta_{0}\cos(2\pi
i/3+\phi(z))].$ (14)

Except for the multiplicative factor in front, Eq. (13) coincides with the
Bloch decomposition along the $z$-direction of the Hamiltonian of electrons
hopping on a two dimensional lattice in the presence of a uniform magnetic
field, if the latter is rendered in the Landau gauge and the phason $\phi$ is
identified with the $k_{z}$ quasi-momentum. The algorithm (2) used for the
height of the coupling bridges sets the value of the virtual magnetic field to
$\frac{1}{3}$-unit of flux per resonator. This connection enables one to
effortlessly understand the topological character of the dynamics and the
ensuing bulk-boundary correspondence. Specifically, the three surface
dispersion bands seen in Fig. 1c are the three spectral bands seen in the
Hofstadter butterfly at $\frac{1}{3}$-flux [46] and, as such, the spectral
gaps between these bands carry Chern numbers $\pm 1$. This implies that the
$x$-terminated sample will display topological edge modes which disperse with
the variation of the phason. More formal analysis is supplied in the
Supplementary Information.

The spectrum of an entire row of resonators $q$-twisted Floquet boundary
conditions imposed in the $z$-direction is reported in figure 1d as function
of $\phi$ and $q$ and the topological edge modes can be seen as the sheet
colored in purple. Taking a slice at a fixed $q$ reveals precisely one chiral
edge band per edge and the slopes of these bands are consistent with the
values of the Chern numbers (see Supplementary Information). Furthermore,
examination of the eigenfunctions leads to the observation of right edge, bulk
and left edge modes for $\phi=0.6\pi,\,\pi,\,\text{and}\,1.4\pi$ in the top,
middle and bottom panels of Fig. 1e, respectively.

Demonstration of topological surface wave transport. We now focus on the
demonstration of the topological surface wave transport. Experiments are first
conducted on the system shown in Fig 1 (see Methods experimental protocol).
Figure 2a displays the experimentally measured EBE field profile at 42.45 kHz
and the harmonic numerical simulation at 41.88 kHz. In both cases, excitation
is applied using piezoelectric actuators on the bottom supercell. We found the
EBE mode experimentally by examining the mode shapes of all measured resonance
peaks in the spectrum shown in Figure 2b. Vertical oscillation of the field
profile is also observed featuring modal nodes and antinodes, owing to the
$z$-directional dynamical phase. The experimental and numerical results
provide satisfactory agreement. To quantitatively compare the retrieved mode
profile in Fig. 2a with the analytical solution Eq. (12), we apply wavelet
transform and mode decomposition on the numerical mode profile. In detail, we
first divided the cuboid into 9 columns. Then, the wavelet transform technique
is applied to the wave component of each column to determine the corresponding
coefficients (see Supplementary Information). Last, we take average on the
absolute values of these coefficients. The outcome after linear interpolation
is illustrated as a heat map in Fig. 2d. As a reference, a purple curve is
given to provide the $k_{z}$-$\phi$ relation at 41.88 kHz on the cut plane of
the dispersion diagram (Fig 1d). Satisfactory agreement is found between the
eigenmode analysis of the finite lattice and the dispersion diagram. Next, we
adopt mode decomposition on each of the 20 supercells along the $z$-direction
to determine the relative strengths of all modes. Figure 2e illustrates the
corresponding modal coefficients which are normalized with the maximum of
coefficient at respective values of $z$. The bases for mode decomposition are
from the corresponding mass-spring model whose parameters are extracted from
Fig. 1c. Since only 20 supercells are involved in the synthetic dimension, the
stiffness matrix does not evolve strictly adiabatically. As a result, other
bulk modes always coexist. However, the EBE mode, labeled as the 7th mode in
Fig. 2e, is always dominant in terms of modal coefficients of all the
supercells, meaning that length of the synthetic dimension is sufficiently
long to approach adiabatic limit. The consistence of the results from wavelet
transform and mode decomposition analysis validates the correctness of the WKB
solution.

Figure 4: Robust topological surface wave pumping. a Eigenmode of the
defective structure where 3 by 3 unit cells of pillars are removed in the
center of the structure at 41.75 kHz. The defect is constructed by removing
resonators and pillars in the dotted line box b Eigenmode of the disordered
structure where a random normal distribution of errors are added on all
geometric parameters at 41.88 kHz. Figure 5: Topologically protected surface
wave splitter.  a Schematic of surface wave pumping system and the
corresponding phase modulation functions. b The displacement field
distribution of the surface wave splitter. The surface wave is injected at the
center of the bottom edge with the frequency at $f=$41.88 kHz.

We conduct the transient analysis for better showcasing the pumping process.
In particular, the right edge mode $\bm{\varphi}_{n}(0)$ at the bottom
supercell is excited by using a series of piezoelectric patches, each attached
on one side of each resonator (see Methods experimental protocol). The
polarization directions of these piezoelectric patches are identical, while
the applied voltages are distributed as $V_{0}\bm{\varphi}_{n}(0)f_{z}(t)$,
where $V_{0}=1~{}\text{V}$ denotes the voltage amplitude, and $f_{z}(t)$ is a
50-cycle tone burst signal $f_{z}(t)=H(50/f_{c}-t)[1-\cos(2\pi
f_{c}t/50)]\sin(2\pi f_{c}t)$ (top panel of Fig. 3), with $H(t)$ being the
Heaviside function and $f_{c}=41.88$ kHz. Figure 3a-c displays the snapshots
of surface wave propagation at representative time instants. Initially, the
right edge mode is excited on the bottom at $t=0.5\ \text{ms}$ (Fig. 3a). As
time progresses, the wave packet propagates in the synthetic dimension $z$ and
transitions into the bulk mode at $t=2.5\ \text{ms}$ (see Fig. 3b).
Eventually, the left edge mode is well formed on the top of the cuboid at
$t=4\ \text{ms}$ (Fig. 3c). The wave packet will follow the same evolution
path transitioning from the left edge mode back to the right one if the
transient simulation continues. A more detailed demonstration can be found in
Supplementary Movies.

Robustness of topological surface wave transport. Indeed, the geometric
imperfections in sample fabrication are inevitable due to the errors of
millers of CNC machines, as minor discrepancies between simulations and
measurement are visible in Fig. 2a, b. Nevertheless, the topological surface
wave transport is evidently observed, thanks to some intriguing wave transport
characteristics, such as robustness against geometrical impurities or defects.
To illustrate this, Fig. 4a shows a lattice defect constructed by removing
$3\times 3$ pillars in the middle of the structure. The corresponding
eigenfrequency of the EBE mode of the defective cuboid (41.75 kHz) is quite
close to that of a perfect cuboid (41.88 kHz). The resulting spatial profile
shows that the topological pumping behavior survives and the edge modes can be
smoothly pumped from one side to the other despite the large-scale geometric
defect. Moreover, we also consider the influence of geometrical disorders. In
the sample fabrication, the machining error is about $0.02\ \text{mm}$ for our
sample. Therefore, we introduce errors that satisfy a normal distribution
$\mathcal{N}(0\ \text{mm},0.02\ \text{mm})$ to the dimensions of all
resonators, including their lengths, heights and widths. The spatial profile
of the EBE mode with disorders is shown in Fig. 4b. The eigenfrequency of the
EBE mode shifts slightly, and the spatial profile agrees with that of the
perfect lattice in the $x$-direction, indicating the topological pumping is
robust against disorders. However, in the $z$-direction, we see amplitudes of
resonators in the top part are larger than amplitudes of resonators in the
bottom part, which is actually a sign of Anderson localization. Since along
$z$ direction, the displacement field is harmonic, the disorders make the
eigenmode localized at the top. And it is observed in the experiment (see Fig.
2a) that the eigenmode is localized at the top.

Application of topological wave transport as wave splitter. The surface wave
topological pumping is promising for practical applications. To show that, we
design a topological split-flow device which performs robust surface wave
splitting. As shown in Fig. 5a, the splitter is an assembly of two domains
with opposite $\phi$-$z$ distributions, separated by a domain wall (yellow in
Fig. 5a). Specifically, the upper section with 20 supercells of the left
domain is designed with a linear $\phi$ transition from $0.6\pi$ to $1.4\pi$,
whereas that of the right domain is assigned an opposite $\phi$ transition,
i.e. from $1.4\pi$ to $0.6\pi$. As for the lower section with 3 supercells,
$\phi$ keeps constant at $0.6\pi$ and $1.4\pi$ for the left and right domains,
respectively. The excitation is located in the middle of the bottom. Within
the lower section of the surface wave splitter, there exists a localized
interface mode. As the incidence reaches the upper half, due to the opposite
gradients of $\phi$, the interface mode is split into two components, each
following the typical EBE evolution but tracing opposite paths. Thanks to
topological protection, the propagation is immune against back reflection from
the discontinuity of the upper and lower sections. As such, our design, based
on phason engineering and topological pumping, provides an avenue for the
application of elastic surface wave beam splitters and smart patterning. In
addition, our design covers the short-wavelength range such that we have the
opportunity to engineer the dispersion with respect to $k_{z}$ quasi-momentum.
This involves modulations along the vertical direction and opens up a new
dimension in the design space for surface wave, which is yet to be explored.

## Discussion

In conclusion, we have evidenced the topological surface wave transport in
modulated phononic crystals through edge-to-edge topological pumpings
associated with the 2D quantum Hall effects by the physical rendering of
synthetic spaces. These observations imply that the system is characterized by
a non-zero Chern number and therefore the topological pumping is immune to
bulk scattering and exhibits strong protection against design imperfections.
The modulated phononic crystals with synthetic spaces offer a platform and
route for efficient surface wave topological mode transport by engineering
desired patterns on a phason-torus in the finite structure. The phason space
augments the physical space and opens a door to higher-dimensional physics in
acoustics and mechanics. Although we focused on the elastic implementation
using synthetic spaces, our approach can be generalized to other degrees of
freedom, such as additional frequency dimensions can also be harnessed for the
frequency modulation. Going forward, it will be important to develop and
explore such broader connections, as the idea of topological matter in
synthetic dimensions is very general and the extension of this approach to
other complex orbits is much awaited. At last, we emphasize that, in order to
achieve a reasonable adiabatic regime, the number of chains in our
experimental set-ups is appreciable and, whereas this is perfectly fine for
the demonstration purposes, it could be an obstacle for practical
applications. It will be interesting to explore if this strategy can be
deployed for our phononic crystals in order to reduce the number of chains
needed for the topological pumping of surface wave.

## Methods

Sample preparation

The experimental sample made of aluminum, having Young’s module
$E=69~{}\text{GPa}$, Poisson’s ratio $\nu=0.33$, and density
$\rho=2700~{}\text{kg}/\text{m}^{3}$, is fabricated using the computer
numerical control (CNC) milling machine with a manufacturing precision of
$0.02~{}\text{mm}$. It consists of an array of resonators
$(6~{}\text{mm}\times 3.5~{}\text{mm}\times 10~{}\text{mm})$ with the number
of 9 along the $x$-direction and 20 along the z direction, which are
integrated with a cuboid $(150~{}\text{mm}\times 50~{}\text{mm}\times
200~{}\text{mm})$. For convenience, each resonator has an address $(i,j)$. In
the x direction, resonators $(i,j)$ and $(i+1,j)$ are connected by height-
modulated pillars $(4~{}\text{mm}\times 1.5~{}\text{mm}\times h_{ij})$ which
satisfy the protocol $h_{ij}=h_{0}[1+\Delta_{0}\cos(2\pi i/3+\phi_{j})]$,
where $h_{0}=7~{}\text{mm},\Delta_{0}=0.15$ is the average thickness of
horizontal channels. Besides, $\phi_{j}=\phi_{s}+(\phi_{f}-\phi_{s})j/N$,
where $\phi_{s}=0.6\pi,\phi_{f}=1.4\pi,N=20$. In the $z$-direction, all
resonators are connected by pillars of the same size $(2~{}\text{mm}\times
6.5~{}\text{mm}\times 3.8~{}\text{mm})$.

Experimental protocol

In the experiment, the sample is supported by four points to mimic the free
boundary condition. A piezoelectric ceramic patch (PZT) is attached to the
right side of the cuboid to excite the target eigenmode state. A wide spectrum
pseudo-random excitation within the probing ranges from $20-50~{}\text{kHz}$
is generated by a Tektronix AFG3051C arbitrary waveform generator and
amplified by a Krohn-Hite high-voltage power amplifier, which is finally
applied across the PZT source. A 1D scanning laser doppler vibrometer (SLDV,
Polytech PSV-500) is used to measure the vibration velocity of resonators in
the $y$-direction, where high-gain reflective tape is stuck on the surface of
each resonator to enhance the reflection of the laser. The velocity signal
from the vibrometer is further recorded by the PSV-500 data acquisition. Note
that the experiment is repeated and averaged 5 times on each resonator of the
system to filter out part of the noise. The normalized amplitude spectrum
obtained by applying the Fourier transform to the time-domain signals
collected at the resonator $(2,18)$ is shown in Fig. 2c. A series of resonant
peaks are observed in the frequency range. By checking the mode shape of each
resonance peak in the frequency spectrum, the EBE state corresponding to the
frequency at 42.45 kHz is identified. Moreover, a full field measurement at
42.45 kHz is conducted by exciting the system with a 200-cycle sine burst, and
the same EBE state is measured.

Numerical simulations

The full-wave finite-element method simulations in this work are all performed
using the commercial software COMSOL Multiphysics. The material of 3D
structure is implemented by Aluminum [solid] from COMSOL Material Library.
Eigenfrequency analysis within the “Solid Mechanics” is carried out to
calculate the eigenfrequencies and eigenmode of the unit cell, supercell, and
cuboid. The boundary conditions for all the cases are set as free boundary
conditions except for Floquet periodicity boundary conditions of the unit cell
along $x$ and $z$ direction, and of the supercell along $z$ direction. For the
transient analysis in Fig. 3, time-dependent analysis in the ”Solid Mechanics”
are used. Piezoelectric patches (PZT-5H in COMSOL Material Library), are
attached on one side of each resonator of the bottom supercell. The
polarization directions of these piezoelectric patches are identical, while
the applied voltages are distributed as $V\bm{\varphi}_{n}(0)f_{z}(t)$.

###### Acknowledgements.

This work is supported by the Air Force Office of Scientific Research under
Grant No. AF 9550-18-1-0342 and AF 9550-20-1-0279 with Program Manager Dr.
Byung-Lip (Les) Lee, the Army Research Office under Grant No. W911NF-18-1-0031
with Program Manager Dr. Daniel P Cole, and the NSF CMMI under Award No.
1930873. Rui Zhu acknowledges support from the National Natural Science
Foundation of China (NSFC) under Grant No. 11991033. Emil Prodan acknowledges
support from the U.S. National Science Foundation through the grants
DMR-1823800 and CMMI-2131760.

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