Snowflake Topological Insulator for Sound Waves
Christian Brendel
Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
Vittorio Peano
Department of Physics, University of Malta, Msida MSD 2080, Malta
Oskar Painter
Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, USA
Florian Marquardt
Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany and
Institute for Theoretical Physics, University of Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
We show how the snowflake phononic crystal structure, which has been realized experimentally
recently, can be turned into a topological insulator for sound waves. This idea, based purely on
simple geometrical modifications, could be readily implemented on the nanoscale.
Introduction
. – First examples of topologically pro-
tected sound wave transport have just emerged during
the past three years. So far, experimental implementa-
tions exist on the centimeter-scale, both for the case of
time-reversal symmetry broken by external driving [1],
such as in coupled gyroscopes, as well as for the case
without driving [2–5], such as in coupled pendula. More-
over, a multitude of different implementations have been
envisioned theoretically [6–19]. However, it is highly de-
sirable to come up with alternative design ideas that may
be realized on the nanoscale, eventually pushing towards
applications in integrated phononics. The first theoreti-
cal proposal in this direction [20] suggested to exploit the
optomechanical interaction to imprint the optical vor-
ticity of a suitably shaped laser beam to generate chi-
ral sound wave transport in a phononic-photonic crys-
tal. On the other hand, if one wants to avoid the strong
driving by an external field, purely geometrical designs
are called for. One remarkable idea of Mousavi et al.
[21] posited creating a sound wave topological insulator
by designing a phononic crystal structure made from a
material that would be carefully engineered by a pat-
tern of small holes to achieve degeneracy between vi-
brations that are symmetric and antisymmetric to the
plane of the sample. The appearance of a fine-grained
length-scale much smaller than the wavelength, however,
makes it impossible to use this idea all the way down
to wavelengths comparable to the smallest feature sizes
allowed by nanofabrication. In the present manuscript,
we propose a very simple modification to an already ex-
isting structure, the so-called snowflake phononic crys-
tal. The snowflake crystal has already proven to be a
reliable platform for nanoscale optomechanics [22], and
could also support pseudomagnetic fields for sound waves
[23]. With the proposed modification, which is inspired
by an idea first analyzed by Wu and Hu for photonic sys-
tems [24] (see also [16, 17, 25]), one will be able to create
a topological insulator for sound waves based on a proven
nanoscale platform.
The envisaged system consists of snowflake-shaped
holes of alternating sizes, in a periodic arrangement on
a triangular lattice. This snowflake topological insulator
can be viewed as a metamaterial that supports topolog-
ically protected sound waves whose typical wavelength
is larger than the underlying lattice scale. Such elastic
waves propagate along arbitrarily shaped domain walls
engineered by appropriately varying the snowflake size.
We will show that the topological protection is guaran-
teed if locally (at the lattice scale) the point group sym-
metry of the snowflake design is mantained.
Platform
.
– We assume a planar quasi-two-
dimensional phononic crystal slab exhibiting a six-fold
rotational symmetry (
C
6
) as well as a discrete transla-
tional symmetry (
T
a
) on a triangular lattice, with a lat-
tice constant
a
. The most straightforward implemen-
tation consists in the snowflake phononic crystal. This
crystal has been explored before in the context of optome-
chanics [22], since it is both a photonic and a phononic
crystal, although we will only make use of its phononic
properties. Its structure is shown in Fig. 1a.
Symmetries and folding
. – Due to the
C
6
-symmetry,
the acoustic band structure is forced to have Dirac cones
at the two high-symmetry points,
~
K
=
π/a
(4
/
3
,
0)
and
~
K
′
=
π/a
(2
/
3
,
2
/
√
3)
. Now consider a single snowflake-
shaped hole surrounded by six other such holes. Our
aim will be to break the original translational symmetry
by changing the central snowflake in this configuration,
thereby enlarging the real-space unit cell by a factor of
√
3
(Fig. 1a). Conversely, this will reduce the size of the
first Brillouin zone (BZ) by the same factor. To antici-
pate this reduction, we imagine what happens when the
original band structure, obtained for the as-yet unper-
turbed structure, gets folded back into the new BZ (see
Fig. 1a-b). This will map the Dirac cones from
~
K
and
~
K
′
of the old BZ to the
Γ
-point of the new BZ, forming a de-
generate pair of double Dirac cones at
~
Γ = (0
,
0)
(Fig. 1c-
arXiv:1701.06330v1 [cond-mat.mes-hall] 23 Jan 2017