Observation of a phononic quadrupole topological insulator
Marc Serra-Garcia,
1, ∗
Valerio Peri,
1, ∗
Roman S¨usstrunk,
1
Osama R.
Bilal,
1, 2
Tom Larsen,
3
Luis Guillermo Villanueva,
3
and Sebastian D. Huber
1
1
Institute for Theoretical Physics, ETH Zurich, 8093 Z¨urich, Switzerland
2
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
3
Advanced NEMS Group,
´
Ecole Polytechnique F´ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland
(Dated: August 18, 2017)
The modern theory of charge polarization in solids is based on a generalization of Berry’s phase.
Its quantization lies at the heart of our understanding of all systems with topological band structures
that were discovered over the last decades. These include the quantum Hall effect, time-reversal
invariant topological insulators in two and three dimensions as well as Weyl semi-metals. Recently,
the theory of this quantized polarization was extended from the dipole- to higher multipole-moments
[1]. In particular, a two-dimensional quantized quadrupole insulator is predicted to have gapped
yet topological one-dimensional edge-modes, which in turn stabilize zero-dimensional in-gap corner
states. However, such a state of matter has not been observed experimentally. Here, we provide the
first measurements of a phononic quadrupole insulator. We experimentally characterize the bulk,
edge, and corner physics of a mechanical metamaterial and find the predicted gapped edge and
in-gap corner states. We further corroborate our findings by comparing the mechanical properties
of a topologically non-trivial system to samples in other phases predicted by the quadrupole theory.
From an application point of view, these topological corner states are an important stepping stone
on the way to topologically protected wave-guides in higher dimensions and thereby open a new
design path for metamaterials.
A non-vanishing dipole moment p = hΨ|r|Ψi in an
insulator
2,3
does not lead to any charge accumulation in
the bulk. However, it manifests itself through uncom-
pensated surface charges and hence induces potentially
interesting surface physics, see Fig. 1a. The dipole mo-
ment p is expressible through Berry’s phase,
2,4
which
in turn can lead to its quantization.
5–15
All observed
topological insulators fit into this framework of quan-
tized dipole moments,
6
or mathematical generalizations
thereof.
12
Whether higher order moments of the electronic
charge distribution, such as the quadrupole moment, can
lead to distinctly new topological phases of matter re-
mained unclear.
Recently, a theory for a quantized quadrupole insulator
was put forward
1
based on its phenomenology: A bulk
quadrupole moment in a finite two-dimensional sample
gives rise to surface dipole moments on its one-dimensional
edges as well as to uncompensated charges on the zero-
dimensional corners, see Fig. 1b. The former is indicating
gapped edge modes while the latter motivates the pres-
ence of in-gap corner excitations. This also defines the
key technological use of such a quadrupole insulator in
mechanical or optical metamaterials: In two dimensions,
the localized corner modes can be used to sense signals in
the bulk which are then exponentially enhanced towards
the corners, where they can be measured efficiently.
16
In
three dimensions, the corner modes translate into one-
dimensional modes which can be used to shuttle energy
in a topologically protected way
17–20
between two points
in space, useful for quantum information processing.
21
The phenomenology of gapped edges and gapless cor-
ners can be formalized mathematically. Benalcazar et
al.
1
proposed to use nested Wilson loops as a way to ob-
tain a quantized quadrupole moment (see App. A): Wil-
son loop operators depend only on the bulk properties
and encode the edge physics via their eigenvalues ν
±
(k
α
),
α = x, y, known as Wannier bands.
22
If the Wannier bands
ν
±
(k
α
) are gapped, the eigenvectors of the Wilson loops
can be used to define the bulk-induced edge polarization
p
±
α
. In the same way as for the conventional topological
insulators,
7
symmetries are required for the quantization
of p
±
α
. It turns out that non-commuting mirror symme-
tries M
x
and M
y
lead to p
±
α
∈ {0, 1/2}. In particular, the
sought after quantized quadrupole phase is described by
1
(p
±
x
, p
±
y
) = (1/2, 1/2). (1)
As a corner terminates two edges,
p
±
x
, p
±
y
= (1/2, 1/2)
could suggest that each of them supports two in-gap states.
However, it is an important hall-mark of the bulk nature
of the quadrupole insulator that each corner hosts only
one mode, cf. Fig. 1b.
1
A concrete tight-binding model for a two-dimensional
quantized quadrupole insulator is shown in Fig. 1c.
1
The
dimerized hopping with amplitude λ and γ leads to a
band-gap between two pairs of degenerate bands for λ 6=
γ (see App. B). The black (red) lines in Fig. 1c indi-
cate positive (negative) hoppings, effectively emulating a
magnetic π-flux per plaquette. This π-flux requires the
mirror-symmetry around the horizontal axis (M
y
) to be
accompanied by a gauge-transformation, leading to the
non-commutation of M
x
and M
y
. The present model
also has C
4
rotational symmetry (again up to a gauge-
transformation) forcing p
±
x
= p
±
y
as well as particle hole-
symmetry giving rise to the aforementioned double degen-
eracy of the Bloch bands. For γ < λ the topological phase
(p
±
x
, p
±
y
) = (1/2, 1/2), whereas for γ > λ, the trivial phase
(0, 0) is realized.
1
Here, we seek a mechanical implemen-
tation of a quadrupole insulator with ¨x
i
= −D
ij
x
j
, where
arXiv:1708.05015v1 [cond-mat.mtrl-sci] 16 Aug 2017
2
the dynamical matrix D
ij
couples local degrees of freedom
x
i
according to the model in Fig. 1c. Note that in the ab-
sence of a Pauli-principle, the notion of “filled bands” has
to be replaced by “bands below the gap of interest.”
23–25
We implement the quadrupole insulator using the con-
cept of perturbative mechanical metamaterials.
26
The
starting point is a single-crystal silicon plate with dimen-
sions 10×10×0.7 mm, whose mechanical eigenmodes are
described by the displacement field u(r). We work with
the first non-solid-body mode which is characterized by
two perpendicular nodal lines in the out-of-plane compo-
nent of u(r), see Fig. 1d and Fig. 2a. By spectrally sep-
arating this mode from the modes below and above it,
one can describe the dynamics in some frequency range
by specifying only the amplitude x
i
of the mode of in-
terest of a given plate i. The hopping elements in D
ij
are then implemented by thin beams between neighboring
plates. The nodal structure of the mode allows to mediate
couplings of either positive or negative sign, depending on
which sides of the nodal lines are connected by the beams.
λ
γ
a b
c d
10 mm
FIG. 1: Quadrupole topological insulator a, in a finite
size system, a bulk dipole moment induces surface charges as
illustrated by the spheres. b, A bulk quadrupole moment with
its accompanying edge dipoles and corner charges. c, a con-
crete model for a system with a non-vanishing quadrupole mo-
ment. Thin (thick) lines denote weak (strong) hoppings with
strength γ and λ respectively. The red (black) lines indicate a
negative (positive) hopping amplitude resulting in a π-flux per
plaquette. d, metamaterial design implementing the model in
c. The out-of-plane plate-modes with two nodal lines (dashed
white lines) are coupled via the bent beams. Beams connect-
ing different sides of a nodal line (shaded red) mediate negative
coupling matrix elements. The gray areas in c & d mark the
unit cell of the tight-binding model.
Moreover, the distance to the nodal line controls the cou-
pling strength mediated by a given beam. Combinatorial
search
27
followed by a gradient optimization
26
leads to the
design in Fig. 1d. Note that the rounded corners and the
semi-circular exclusions guarantee spectral separation, cf.
Fig. 2a, whereas the holes in the center adjust for plate-
dependent frequency shifts induced by the beams. The
beams are kinked to decrease their longitudinal stiffness
ensuring spectral separation.
All measurements shown are performed using the same
scheme: The plates are excited with an ultrasound air-
transducer. The transducer has a diameter of 5 mm and
is in close proximity to the sample, such that only a single
plate is excited. We measure the response of the excited
plate with a laser-interferometer. In this way, we measure
the out-of-plane vibration amplitude ∆z
i
∝ ψ
2
i
, where ψ
i
is the eigenmode at the measured frequency. The inset of
Fig. 2a shows the local mode of a single plate measured
in this way. In all other figures we display the mechanical
energy ε
i
∝ ∆z
2
i
.
To identify the in-gap states we take a measurement of
ε
i
(ν) as a function of frequency ν on all plates i. We then
apply the filters ε
α
(ν) =
P
i
ε
i
(ν)F
i,α
shown in Fig. 2d
to separate the response of the bulk, edges, and corners.
Figs. 2b & e show the resulting spectra for two differ-
ent samples (see App. C). In the topologically trivial case
with γ > λ, one can observe two frequency bands where
the system absorbs energy (the theoretically predicted lo-
cation of the bands is indicated in gray). Two features
characterize this trivial phase: (i) No frequency range is
dominated by the edge or corner response. Moreover, the
relative weight of the three curves is in good accordance
with the respective number of sites in the bulk, edges, and
corners, respectively. (ii) No resonances appear in the gap
between 36.0 kHz and 36.7 kHz. For the sample with γ < λ
in Fig. 2e, two key-features of the quantized quadrupole
phase appear: (ii) close to 36.0 kHz and 36.7 kHz, the re-
sponse is dominated by the edges, indicative of the bulk-
induced gapped edge modes. (ii) Sharp resonances at the
corners appear in the gap region. A small mirror sym-
metry breaking leads to the non-degeneracy of the in-gap
states which we discuss below.
The spectra in Fig. 2b & e allow to identify three fre-
quency regions B, E, and C, where the bulk (blue), edge
(orange), or corner (green) response dominates. To es-
tablish the quadrupole nature of the metamaterial, we
analyze the site-dependent, frequency integrated response
ε
α
i
=
P
ν∈α
ε
i
(ν) with α = B, E, C. In Fig. 3a–c we show
the resulting spatial profiles. Note that the bulk induces
gapped edge-modes on all four sides of the sample.
The hallmark of the quadrupole phase lies in the count-
ing of corner modes: Each corner terminates two-gapped
edges, nevertheless, they all host only one in-gap mode.
1
In Fig. 3d, we show the response ε(ν) for the four corner
plates. The resonances in the four plates are split by the
imperfect termination and the disorder induced by the fab-
rication. However, each corner hosts only one resonance
peak. Given the imperfection of our setup, where disor-
3
0 20 40 60
Frequency [kHz]
10
−1
10
0
10
1
10
2
Energy [a.u.]
−1
1
∆z [nm]
−1
1
∆z [nm]
35.0 35.5 36.0 36.5 37.0
Frequency [kHz]
0.0
0.2
0.4
0.6
0.8
1.0
Energy [a.u]
bulk
edge
corner
x
y
0.2
0.4
0.6
0.8
1.0
35.0 35.5 36.0 36.5 37.0
Frequency [kHz]
0.0
0.2
0.4
0.6
0.8
1.0
Energy [a.u]
a b
ec d
FIG. 2: Quadrupole in-gap states a, Spectrum on a single plate indicating the large separation between the targeted mode
around 36 kHz (shaded gray) and the bands above and below. The left inset shows the mode profile measured on a single plate
(the black dots mark the measurement points used for the interpolation), whereas the right inset shows the numerically calculated
mode profile. In d, the response of all plates at an arbitrary frequency (37.0 kHz) is shown. These images are then multiplied by
the displayed filters to determine bulk, edge, and corner response. b and e show the resulting spectra for the trivial and non-trivial
sample, respectively. For the trivial case, one can see two bands (the gray area indicates the theoretically predicated location
of the two bands) and a central gap with no resonances. For all frequencies, the weights of the bulk (blue), edge (orange), and
corner (green) responses follow their fraction of the total 10x10 system. The non-trivial case (e) shows bulk- and edge-dominated
frequency-regions and strong corner peaks in the middle of the gap. c, Photo of the setup.
der would generically split multiple mode per plate, this is
only compatible with the mode-counting of a quadrupole
insulator.
To corroborate our claim of observing a quadrupole in-
sulator, we further explore the phase diagram of Ref. 1.
When the C
4
-symmetry is broken by allowing for differ-
ent hoppings in x- and y-direction (see App. B), the phase
(p
±
x
, p
±
y
) = (1/2, 0) can be reached via a gap-closing of
the surface modes. The (1/2, 0)-phase is characterized by
gapped edge spectra on two parallel edges and no emer-
gent edge physics on the perpendicular surfaces.
1
More-
over, the induced edge modes are in a trivial state and no
corner charges are induced. In Fig. 3e, we show measure-
ments on a sample in the (1/2, 0)-phase, where no in-gap
states appear and the frequency region dominated by the
edges draws its weight from only two surfaces.
In addition to the experimental data presented above,
we also validate our system through extensive numerical
calculations. The design process for the sample shown
in Fig. 1d requires a finite-element simulation of the dis-
placement fields u
i
(r) on four unit cells containing a total
of 16 sites i. The modes obtained in this way can then be
projected onto the basis of uncoupled plate-modes u
0
i
(r).
In this way a reduced order model
˜
D
ij
in the frequency
range of the modes u
0
i
(r) is obtained.
26
In Fig. 4a, we
show the resulting model extended to a 10 ×10 system.
The nearest neighbor couplings indeed follow the blueprint
of the target model shown in Fig. 1d. However, spurious
long-range couplings mediated by off-resonant admixing
of other single-plate modes induce a certain amount of
mirror-symmetry breaking. This is most notable in the
y-direction, where negative next-to-nearest neighbor cou-
plings are mapped to positive ones, which is not corrected
for in the gauge-transformation in M
y
.
The reduced order model
˜
D
ij
can also be used to calcu-
late the topological indices (p
±
x
, p
±
y
). The gapped Wannier
bands ν
±
x
(k
y
) and ν
±
y
(k
x
) are shown in Fig. 4b. Note that
the M
x
symmetry implies ν
+
x
(k
y
) + ν
−
x
(k
y
) = 1/2 and the
same for x ↔ y.
1
The absence of an exact M
y
symme-
try indeed leads to a breaking of this rule. This is also
reflected in the value of the polarizations
(p
+
x
, p
−
x
) = (0.50, 0.49), (2)
(p
+
y
, p
−
y
) = (0.58, 0.56). (3)
As expected from the structure of
˜
D
ij
shown in Fig. 4a,
the polarizations are not precisely quantized. However,
the phenomenology of in-gap corner modes is still observed
as the symmetry breaking terms do not lead to any gap-
closing, neither on the edge nor in the bulk.
The results presented in this paper underline the power
of perturbative metamaterials.
26
On one hand, we lever-
aged this technique to find a first implementation of a
quantized quadrupole insulator, a new class of topological
4
0 2
4
6 8
x
0
2
4
6
8
y
0
1
weights
0 2
4
6 8
x
0
2
4
6
8
y
0
1
weights
0 2
4
6 8
x
0
2
4
6
8
y
0
1
weights
0
1
0
1
0
1
Energy [a.u.]
35 36 37
Frequency [kHz]
0
1
0 2
4
6 8
0
2
4
6
8
0
1
35.0 35.5 37.0
Frequency [kHz]
0
1
Energy
bulk
edge
corner
a b
c d
e
FIG. 3: Edge and corner modes a – c, Normalized inte-
grated weights of the response of frequency regions in Fig. 1e
where bulk (a), edge (b), and corner modes (c) dominate. d,
Spectral response of the four corner sites in clock-wise arrange-
ment starting from the top left corner. The combination of
gapped edge modes on all four edges, see b, together with the
single edge mode per site evidences the quadrupole nature of
our metamaterial. e, Spectrum and edge dominated modes
of a system in the non-quadrupole phase (p
x
, p
y
) = (1/2, 0)
showing no corner states but surface modes on two of the four
edges.
materials. On the other hand, the platform of a continu-
ous elastic medium provides a direct route to technological
applications for any theoretical idea which can be repre-
sented by a tight-binding model.
Acknowledgments
We acknowledge financial support from the Swiss Na-
tional Science Foundation and the NCCR QSIT.
Appendix A: Topological quantum number: Nested
Wilson loops
Assuming two bands n = 1, 2 are filled, one can use the
non-abelian Berry phase A
x
nm
(k) = ihu
m
(k)|∂
k
x
|u
n
(k)i of
the Bloch wave-functions |u
n
(k)i to construct the Wilson-
0 2
k/π
0.0
0.5
1.0
Wannier bands
ν
−
x
ν
+
x
ν
−
y
ν
+
y
a b
FIG. 4: Reduced model and Wannier bands a, Extracted
reduced model for our design. Black (red) lines indicate posi-
tive (negative) couplings between the plate modes, whereas the
thickness of the lines encodes the hopping amplitude. The un-
wanted next-to-nearest neighbor couplings arise from second-
order effects involving other plate modes and break the M
x
and
M
y
symmetries. b, Calculated Wannier bands from the model
on the left.
loop operators
W
x
(k
y
) = T exp
i
I
dk
x
A
x
nm
(k)
. (A1)
Here, T denotes the path ordering along a closed loop
in the Brillouin zone. The eigenvalues ν
±
(k
y
) of W
x
(k
y
)
are in one-to-one correspondence to the spectrum of an
edge perpendicular to the x-coordinate
22
(or perpendic-
ular to y when x and y are interchanged). If the edge
modes are gapped, the eigenvectors v
±
n
(k
y
) of W
x
(k
y
) can
be used to split the filled bands in a well-defined way:
|w
±
(k)i =
P
2
n=1
v
±
n
(k
y
)|u
n
(k)i. The nested polarization
is then defined as
p
±
y
=
1
(2π)
2
Z
dk A
y
±
(k), (A2)
with A
y
±
(k) = ihw
±
(k)|∂
k
y
|w
±
(k)i. It can be shown that
the presences of two mirror-symmetries M
x
and M
y
that
do not commute are a necessary requirement for the nested
polarizations p
±
x
and p
±
y
to be quantized to 0 or 1/2.
1
Appendix B: Model
The model shown in Fig. 1 can be expressed with the
help of Γ-matrices Γ
k
= −τ
2
σ
k
, Γ
4
= τ
1
σ
0
, k = 1, 2, 3; τ, σ
are the standard Pauli-matrices. Using these matrices we
can write
1
D(k
x
, k
y
) = [γ
x
+ λ
x
cos(k
x
)]Γ
4
+ λ
x
sin(k
x
)Γ
3
+ [γ
y
+ λ
y
cos(k
y
)]Γ
2
+ λ
y
sin(k
y
)Γ
1
=
4
X
i=1
d
i
(k)Γ
i
.
(B1)
The C
4
-symmetric version of Fig. 1 is obtained by set-
ting λ
x
= λ
y
and γ
x
= γ
y
. The mirror symmetries
5
10
−1
10
0
10
1
γ
x
/λ
x
10
−1
10
0
10
1
γ
y
/λ
y
A B
C
D
(1/2,1/2) (1/2,0)
(0,0)(0,1/2)
A B
C
D
0.0
0.5
1.0
x-Wannier bands
A B
C
D
Path through phase diagram
0.0
0.5
1.0
y-Wannier bands
a b
FIG. 5: Phase diagram. a, Phase diagram of model (B1). The brown area marks the quantized quadrupole phase, whereas the
orange areas are the (1/2, 0) and (0, 1/2) phases with no corner modes but emergent edge physics along two parallel edges. The
dashed lines indictes the C
4
-symmetric line, where the bulk gap is closing at the phase transition. The transitions away from the
C
4
-symmetric line happen through bulk-induced edge-transitions, where no bulk gap is closing. b, The evolution of the Wannier
bands in x and y direction along the path shown in a. The transition from the quadrupole phase to the (1/2, 0) phase is marked
by a gap-closing at 1/2, removing any polarization in the system. The second transition is induced by a gap closing at 0.
are represented by D(−k
x
, k
y
) = m
x
D(k
x
, k
y
)m
†
x
and
D(k
x
, −k
y
) = m
y
D(k
x
, k
y
)m
†
y
with m
x
= τ
1
σ
3
and m
y
=
τ
1
σ
1
, respectively. The eigenvalues of D(k
x
, k
y
) are given
by ζ = ±|d(k)|, leading to two doubly-degenerate bands.
Bulk gap-closings occur when d(k) = 0, which only hap-
pens for the C
4
-symmetric case at λ = ±γ. The spectrum
of the mechanical system is given by ν =
p
ν
2
0
+ ζ, with
a frequency offset ν
0
. Finally, the eigenvectors |u
n
(k)i of
D(k
x
, k
y
) can be used to calculate the Wilson loop oper-
ators of Eq. (A1). The phase diagram and the evolution
of the Wannier bands of model (B1) are shown in the ex-
tended data Fig. 5.
Appendix C: Sample design and preparation
The plate and beam geometry of Fig. 1d implement the
sought after weak and strong, positive and negative cou-
pling matrix elements. The definition of γ as the hopping
strength inside a unit cell and λ between unit cells renders
γ < λ the non-trivial phase. Connected to this identifica-
tion is the notion of how we are allowed to terminate the
system: Surfaces have to be compatible with the unit-cells,
i.e., are not allowed to cut through unit-cells. In turn, this
also means we can use the same design of Fig. 1d and re-
alize all phases shown in this paper by starting from a 10
×10 sample in the (1/2, 1/2)-phase, then move the cut in
y-direction by one row of sites to reach the (1/2, 0)-phase.
Finally we move the termination one column and end up
in the (0, 0)-phase. The coupling matrix element are given
by the ratio of the effective mass-density ρ
eff
of the mode
we use and the beam stiffness connecting two plates. We
use a 700 µm thick Si-wafer in [100] orientation, where we
align the x- and y-axis of our model with the in-plane crys-
talline axes. The mass density of Si is ρ = 2330 kg/m
3
,
the Young’s moduli E
x
= E
y
= E
z
= 130 GPa, the Pois-
son ratios ν
yz
= ν
zx
= ν
xy
= 0.28, and the shear moduli
G
yz
= G
zx
= G
xy
= 79.6 GPa.
28
This results in an off-
set frequency for our mode of ν
0
= 36.716 ± 0.03 kHz and
the coupling matrix elements are given by λ = 1.26 ±
0.03 × 10
9
(rad/s)
2
and γ = 0.43 ± 0.01 × 10
9
(rad/s)
2
.
We produce our sample by laser-cutting with an accu-
racy of ∼ 50 µm, giving rise to the indicated uncertain-
ties (disorder) in the coupling strengths. The wafers are
clamped between two steel plates (each of 3 mm thickness),
cf. Fig. 2c. The impedance miss-match between the steel
plates and the wafer leads essentially to fixed boundary
conditions ∆z = 0. The ultrasound air-transducer used
was SMATR300H19XDA from Steiner & Martins Inc.
Appendix D: Signal analysis
Under the assumption that all modes have the same
quality factor Q ≈ 1000 (determined from the width of
the corner modes), the completness of eigenmodes requires
the integral
R
dν ∆z
i
(ν) ∝
R
dν ψ
2
i
(ν) to be the same for
all sites i. We use this assumption to normalize all our
