week ending

PRL 114, 114301 (2015) PHYSICAL REVIEW LETTERS 20 MARCH 2015

Topological Acoustics
Zhaoju Yang,1 Fei Gao,1 Xihang Shi,1 Xiao Lin,1 Zhen Gao,1 Yidong Chong,1,2 and Baile Zhang1,2,*
1
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371, Singapore
2
Centre for Disruptive Photonic Technologies, Nanyang Technological University,
Singapore 637371, Singapore
(Received 22 December 2014; published 20 March 2015)
The manipulation of acoustic wave propagation in fluids has numerous applications, including some in
everyday life. Acoustic technologies frequently develop in tandem with optics, using shared concepts such
as waveguiding and metamedia. It is thus noteworthy that an entirely novel class of electromagnetic waves,
known as “topological edge states,” has recently been demonstrated. These are inspired by the electronic
edge states occurring in topological insulators, and possess a striking and technologically promising
property: the ability to travel in a single direction along a surface without backscattering, regardless of the
existence of defects or disorder. Here, we develop an analogous theory of topological fluid acoustics, and
propose a scheme for realizing topological edge states in an acoustic structure containing circulating fluids.
The phenomenon of disorder-free one-way sound propagation, which does not occur in ordinary acoustic
devices, may have novel applications for acoustic isolators, modulators, and transducers.

DOI: 10.1103/PhysRevLett.114.114301 PACS numbers: 43.20.+g, 73.43.−f

Since the 1980s, it has been known that the bands of possessed by the topological edge states we will develop.
certain insulators are “topologically nontrivial,” i.e., not We utilize the design concept by incorporating circulating
smoothly deformable into the bands of a conventional fluid elements into a PC structure. As shall be seen,
insulator. Such systems, collectively referred to as the resulting acoustic band structure is topologically non-
“topological insulators” [1–3], can exhibit edge states that trivial, and maps theoretically onto an integer quantum
propagate in a single direction along the edge of a Hall gas [1]—the simplest version of a two-dimensional
two-dimensional sample. These states are “topologically (2D) topological insulator.
protected,” meaning that they are tied to the topology of We note also that several authors have studied topologi-
the underlying bands and cannot be eliminated by pertur- cal vibrational modes in mechanical lattices [17–19].
bations, and are hence immune to backscattering from The present system, by contrast, involves sound waves
disorder or shape variations. Some years ago, Haldane in continuous fluid media, which is considerably more
and Raghu [4] predicted that a similar phenomenon can relevant for existing acoustic technologies.
arise in the context of classical electromagnetism [4–13], A schematic of the proposed system is shown in
which was subsequently borne out by experiments on Fig. 1(a). It is a triangular lattice of lattice constant a,
microwave-scale magneto-optic photonic crystals [6] and where each unit cell consists of a rigid solid cylinder (e.g., a
other photonic devices [8,10,11]. metal cylinder) with radius r1 , surrounded by a cylindrical
In order to realize topological edge states using sound, fluid-filled region of radius r2 . The remainder of the unit
we begin with a spatially periodic medium (in order to have cell (i.e., the region of r > r2 ) consists of a stationary fluid,
band structures), and introduce a mechanism that breaks separated from the fluid in the cylindrical region (i.e., the
time-reversal symmetry (to allow for one-way propaga- region of r1 < r < r2 ) by a thin impedance-matched layer
tion). A periodic acoustic medium, sometimes called a at radius r2 . (This layer can be achieved using a thin sheet
“phononic crystal” (PC) [14], is commonly realized by of solid material that is permeable to sound.) The central
engineering a structure whose acoustic properties (elastic cylinder rotates along its axis with angular speed Ω, which
moduli and/or mass density) vary periodically on a scale produces a circulatory flow in the surrounding fluid in the
comparable to the acoustic wavelength. As for T breaking, region of r1 < r < r2 . (We will not consider the possibility
although traditional acoustic devices lack an efficient of Taylor vortex formation [20] caused by large Ω in
mechanism for accomplishing this, a recent breakthrough experiments because we here focus on a 2D model and the
[15] has shown that strong T breaking can be achieved Taylor vortex does not contribute an effective flux through
in a “meta-atom” containing a ring of circulating fluid. the xy plane.) We assume that the fluid velocity is much
Although these developments have direct device applica- slower than the speed of sound (Mach number of less
tions as acoustic diodes [16] and acoustic circulators [15], than 0.3). The motion of the fluid can be described by a
they do not have the topological protection against defects circulating “Couette flow” distribution [20]: the velocity

0031-9007=15=114(11)=114301(4) 114301-1 © 2015 American Physical Society

 week ending
PRL 114, 114301 (2015) PHYSICAL REVIEW LETTERS 20 MARCH 2015

and heat flow are negligible, the waves obey a “sound

master equation”

1 1
∇ · ρ∇ϕ − ð∂ t þ v~ 0 · ∇Þ 2 ð∂ t þ v~ 0 · ∇Þϕ ¼ 0; ð1Þ
ρ c

where ρ is the fluid density, c is the speed of sound, and

v~ 0 is the background fluid velocity (i.e., the Couette flow
distribution in the region of r1 < r < r2 and stationary
fluid in the region of r > r2 , where r is measured
from the center of each unit cell). The relation between
the velocity potential ϕ and the sound pressure p is
p ¼ ρð∂ t þ v~ 0 · ∇Þϕ. We model the surface of each cylinder
as an impenetrable hard boundary by setting n~ · ∇ϕ ¼ 0,
where n~ is the surface normal vector. We restrict our
attention to time-harmonic solutions with frequency ω
and neglect second order terms as j~v0 =cj2 ≪ 1. With a
pffiffiffi
change of variables Ψ ¼ ρϕ the master equation can be
rewritten as
2
½ð∇ − iA
~ eff Þ þ Vðx; yÞΨ ¼ 0; ð2Þ

where the effective vector and scalar potentials are

~ eff ¼ − ω~v0 ðx; yÞ ;

A ð3Þ
c2
1 1 ω2
Vðx; yÞ ¼ − j∇ ln ρj2 − ∇2 ln ρ þ 2 : ð4Þ
4 2 c
Evidently, Eq. (2) maps onto the Schrödinger equation for a
spinless charged quantum particle in nonuniform vector and
scalar potentials. For nonzero Ω, the inner boundary of the
FIG. 1 (color online). A two-dimensional acoustic topological Couette flow contributes positive effective magnetic flux,
insulator and its band structure. (a) Triangular acoustic lattice and the rest of the Couette flow contributes negative
with lattice constant a. a ¼ 0.2 m in the following calculation.
effective magnetic flux; the net magnetic flux, integrated
Inset: unit cell containing a central metal rod of radius r1 ¼ 0.2a,
surrounded by an anticlockwise circulating fluid flow (flow over the entire unit cell, is zero. The acoustic system thus
direction indicated by red arrows) in a cylinder region of radius behaves like a “zero field quantum Hall” system [23] and is
r2 ¼ 0.4a. (b) Band structures of the acoustic lattice without the periodic in the unit cell.
circulating fluid flow (red curves, Ω ¼ 2π × 0 rad=s) and with It is worth mentioning that a similar approach to
fluid flow (blue curves, Ω ¼ 2π × 400 rad=s). In the gapped construct an effective magnetic vector potential for classical
band structure, the bands have Chern number 1 (blue labels). wave propagation has been discussed by Berry and
Left inset: enlarged view of Dirac cone. Right lower inset: the colleagues [24,25]. These authors showed that an irrota-
first Brillouin zone. (c) Frequency splitting as a function of the tional (“bathtub”) fluid vortex exhibits a classical wave
angular velocity of the cylinder in each unit cell. The degeneracy front dislocation effect, analogous to the Aharanov-Bohm
at the Dirac point with frequency ω0 ¼ 0.577 × 2πca =a (992 Hz) effect. Here, we advance this insight by applying the flow
is removed for Ω ≠ 0.
model to a PC context, so that the effective magnetic vector
potential gives rise to a topologically nontrivial acoustic
field points in the azimuthal direction, with component band structure.
vθ ¼ −½Ωr21 =ðr22 − r21 Þr þ ½Ωr21 r22 =ðr22 − r21 Þð1=rÞ, where From Eq. (1), we can calculate the acoustic band
r is measured from the origin at the axis of the cylinder. structures using the finite element method. For simplicity,
This angular velocity is equal to Ω at radius r ¼ r1 , and we assume the fluids involved are air. The results, with
zero at radius r ¼ r2 . Ω ¼ 0 and Ω ≠ 0, are shown in Fig. 1(b) (the lattice
The propagation of sound waves in the presence of such constant a is set as 0.2 m). For Ω ¼ 0 [red curves in
a steady-state nonhomogenous velocity background is Fig. 1(b)], the acoustic band structure exhibits a pair of
described in Refs. [21,22]. Assuming that the viscosity Dirac points at the corner of the hexagonal Brillouin zone,

114301-2
 week ending
PRL 114, 114301 (2015) PHYSICAL REVIEW LETTERS 20 MARCH 2015

at frequency ω0 ¼ 0.577 × 2πca =a (992 Hz), where ca is

the sound velocity in air.
For Ω ≠ 0 the circulating air flow produces a dramatic
change in the band structure [blue curves in Fig. 1(b)].
Here, we set the angular velocity of the inner rods to be
Ω ¼ 2π × 400 rad=s (achievable with miniature electric
motors). The Dirac point degeneracies are lifted, producing
a finite complete band gap. The frequency splitting at the
zone corners as a function of Ω is plotted in Fig. 1(c).
The ratio of the operating frequency to the band gap, which
is an estimate for the penetration depth of the topological
edge states in units of the lattice constant, is on the order of
ω=δω ≈ 10 for the range of angular velocities plotted here.
For Ω ¼ 2π × 400 rad=s, the band gap ranges from 914 to
1029 Hz, corresponding to a relatively narrow bandwidth
of 115 Hz.
Each acoustic band can be characterized by a topological
invariant, the Chern number [4]. The Berry connection
and Chern number of the nth acoustic band can be defined
as follows:
FIG. 2 (color online). Acoustic one-way edge states.
~ n ¼ ihϕnk j∇~ jϕnk i;
A ð5Þ (a) Dispersion of the one-way acoustic edge states (red curves)
k
occurring in a finite strip of the acoustic lattice, for
ZZ Ω ¼ 2π × 400 rad=s. The left and right red curves correspond
1 ~ n: to edge states localized at the bottom and top of the strip. (b),(c)
Cn ¼ ðdka ∧ dkb Þ∇k~ × A ð6Þ
2π BZ The normalized acoustic pressure p for a left-propagating
acoustic edge state at frequency ω0 ¼ 0.577 × 2πca =a (992 Hz)
for Ω ¼ 2π × 400 rad=s (b) and Ω ¼ 2π × 200 rad=s (c). Lattice
We have numerically verified that the two bands in
parameters are the same as in Fig. 1.
Fig. 1(b), split by the T breaking, have Chern numbers
of 1. The principle of “bulk-edge correspondence” then
predicts that, for a finite PC, the gap between these two edge state with a longer penetration depth because of a
bands is spanned by unidirectional acoustic edge states, narrower band gap.
analogous to the electronic edge states occurring in the Because of the lack of backward-propagating edge modes,
quantum Hall effect [26]. the presence of disorder cannot cause backscattering.
To confirm the existence of these topologically protected Figure 3(a) shows an acoustic cavity located along the
acoustic edge states, we numerically calculate the band interface; the incident wave flows through the cavity, and
structure for a 20 × 1 supercell [27] (a ribbon that is 20 unit excites localized resonances within the cavity, but does not
cells wide in the y direction and infinite along the backscatter. Figure 3(b) shows a Z-shape bend connecting
x direction). As shown in Fig. 2(a), for Ω¼2π ×400rad=s two parallel surfaces at different y; again, the acoustic edge
the band gap contains two sets of edge states, which states are fully transmitted across the bend. Finally, Fig. 3(c)
are confined to opposite edges of the ribbon and have shows a 180-deg bend, which allows acoustic edge states to
opposite group velocities. be guided from the top of a sample to the bottom of the
Figures 2(b) and 2(c) show the propagation of these edge sample; note that the left boundary in this sample is a zigzag
states in a finite (34 × 14) lattice. In these simulations, the boundary, which supports one-way edge states with different
upper edge of the PC is enclosed by a sound-impermeable dispersion relations.
hard boundary (e.g., a flat metal surface), in order to Our proposed system should be quite practical to realize.
prevent sound waves from leaking into the upper half Similar effects can be achieved with alternative designs
space; absorbing boundary conditions are applied to the featuring circulatory fluid velocity distributions: e.g.,
sides. A point sound source with midgap frequency ω0 is having azimuthally directed fans in each unit cell [15] or
placed near the upper boundary. For Ω ¼ 2π × 400 rad=s, stirring with a rotating disc on the top plate [28]. The effect
this excites a unidirectional edge state that propagates to the could be largely tunable from audible to even ultrasonic
left along the interface [Fig. 2(b)]. If the sign of angular frequencies by appropriately scaling down the lattice
velocity were reversed, the edge state would be directed to constant or practically operating at higher band gaps with
the right (not plotted). The field distribution for a reduced a larger Chern number. Acoustic devices based on these
angular velocity Ω ¼ 2π × 200 rad=s [Fig. 2(c)] shows an topological properties can be useful for invisibility from

114301-3
 week ending
PRL 114, 114301 (2015) PHYSICAL REVIEW LETTERS 20 MARCH 2015

[5] Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljacic, Phys.

Rev. Lett. 100, 013905 (2008).
[6] Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljacic,
Nature (London) 461, 772 (2009).
[7] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor,
Nat. Phys. 7, 907 (2011).
[8] Y. Poo, R. X. Wu, Z. F. Lin, Y. Yang, and C. T. Chan, Phys.
Rev. Lett. 106, 093903 (2011).
[9] K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782
(2012).
[10] M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor,
Nat. Photonics 7, 1001 (2013).
[11] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D.
Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit,
Nature (London) 496, 196 (2013).
[12] A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian,
A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233
FIG. 3 (color online). Demonstration of the robustness of (2012).
acoustic one-way edge states against disorder. Topological [13] L. Lu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics
protection requires the waves to be fully transmitted through 8, 821 (2014).
an acoustic cavity (a), a Z-shape bend along the interface (b), and [14] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B.
a 180-deg bend (c). The operating frequency is ω0 ¼ 0.577 × Djafari-Rouhani, Phys. Rev. Lett. 71, 2022 (1993).
2πca =a (992 Hz) and Ω ¼ 2π × 400 rad=s. Lattice parameters [15] R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and
are the same as in Fig. 1. A. Alu, Science 343, 516 (2014).
[16] B. Liang, X. S. Guo, J. Tu, D. Zhang, and J. C. Cheng,
Nat. Mater. 9, 989 (2010).
sonar detection, one-way signal processing regardless of [17] E. Prodan and C. Prodan, Phys. Rev. Lett. 103, 248101
disorders, and environmental noise control, which will (2009).
greatly broaden our interest in military, medical, and [18] C. L. Kane and T. C. Lubensky, Nat. Phys. 10, 39
industrial applications. (2013).
[19] J. Paulose, B. G. Chen, and V. Vitelli, arXiv:1406.3323.
This work was sponsored by Nanyang Technological [20] P. K. Kundu and I. M. Cohen, Fluid Mechanics (Elsevier,
University under start-up grants, and by the Singapore New York, 2012).
Ministry of Education under Grants No. Tier 1 RG27/12 [21] A. D. Pierce, J. Acoust. Soc. Am. 87, 2292 (1990).
and No. MOE2011-T3-1-005. C. Y. D. acknowledges sup- [22] L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered
port from the Singapore National Research Foundation Media II: Point Sources and Bounded Beams (Springer,
Berlin, 1999).
under Grant No. NRFF2012-02.
[23] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
[24] M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and
J. C. Walmsley, Eur. J. Phys. 1, 154 (1980).
*
To whom all correspondence should be addressed. [25] U. Leonhardt and T. Philbin, Geometry and Light: The
blzhang@ntu.edu.sg Science of Invisibility (Dover, New York, 2010).
[1] K. V. Klitzing, Rev. Mod. Phys. 58, 519 (1986). [26] Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
[2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [27] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and
[3] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). R. D. Meade, Photonic Crystals: Molding the Flow of
[4] F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, Light (Princeton University, Princeton, NJ, 2008).
013904 (2008). [28] M. Fink, Phys. Today 50, No. 3, 34 (1997).

114301-4