Inspired by the quantum valley Hall effect, a mechanical topological insulator (TI) purposely built for reconfigurability is proposed and experimentally demonstrated. An aluminum plate serves as the host medium with periodically arranged voids and fixed inclusions used to break mirror symmetry. Reconfigurability is derived from the ability to easily alter the imperfection type (void or fixed inclusion) in any unit cell. The corresponding band structure of the proposed hexagonal unit cell is obtained using numerical means, which documents double-folded Dirac cones at the K-points. The breaking of mirror symmetry results in a topologically protected bandgap. Furthermore, topologically protected edge states (TPES) at the interface of two structures with opposite Chern numbers have been demonstrated numerically, and verified experimentally, for different desired trajectories. These TPES are robust against backscattering at defect locations and sharp bends. The proposed reconfigurable TI can be a stepping-stone platform toward building mechanical logic and circuits, which have advantages over electronic equivalents in harsh operating conditions, or to replace wireless systems near dead-zones of metallic and carbon fiber structures.

1.
L.
Lu
,
J. D.
Joannopoulos
, and
M.
Soljačić
, “
Topological photonics
,”
Nat. Photon.
8
(
11
),
821
(
2014
).
2.
M. Z.
Hasan
and
C. L.
Kane
, “
Colloquium: Topological insulators
,”
Rev. Mod. Phys.
82
(
4
),
3045
(
2010
).
3.
X.-L.
Qi
and
S.-C.
Zhang
, “
Topological insulators and superconductors
,”
Rev. Mod. Phys.
83
(
4
),
1057
(
2011
).
4.
K.
Von Klitzing
, “
The quantized Hall effect
,”
Rev. Mod. Phys.
58
(
3
),
519
(
1986
).
5.
C. L.
Kane
and
E. J.
Mele
, “
Quantum spin hall effect in graphene
,”
Phys. Rev. Lett.
95
(
22
),
226801
(
2005
).
6.
F.
Haldane
and
S.
Raghu
, “
Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry
,”
Phys. Rev. Lett.
100
(
1
),
013904
(
2008
).
7.
Z.
Wang
,
Y.
Chong
,
J. D.
Joannopoulos
, and
M.
Soljačić
, “
Reflection-free one-way edge modes in a gyromagnetic photonic crystal
,”
Phys. Rev. Lett.
100
(
1
),
013905
(
2008
).
8.
A. B.
Khanikaev
,
S. H.
Mousavi
,
W.-K.
Tse
,
M.
Kargarian
,
A. H.
MacDonald
, and
G.
Shvets
, “
Photonic topological insulators
,”
Nat. Mater.
12
(
3
),
233
(
2013
).
9.
M.
Hafezi
,
S.
Mittal
,
J.
Fan
,
A.
Migdall
, and
J.
Taylor
, “
Imaging topological edge states in silicon photonics
,”
Nat. Photon.
7
(
12
),
1001
(
2013
).
10.
V.
Vitelli
,
N.
Upadhyaya
, and
B. G.-g.
Chen
, “
Topological mechanisms as classical spinor fields
,” arXiv:1407.2890 (
2014
).
11.
S. H.
Mousavi
,
A. B.
Khanikaev
, and
Z.
Wang
, “
Topologically protected elastic waves in phononic metamaterials
,”
Nat. Commun.
6
,
8682
(
2015
).
12.
R.
Süsstrunk
and
S. D.
Huber
, “
Observation of phononic helical edge states in a mechanical topological insulator
,”
Science
349
(
6243
),
47
50
(
2015
).
13.
R.
Fleury
,
D. L.
Sounas
,
C. F.
Sieck
,
M. R.
Haberman
, and
A.
Alù
, “
Sound isolation and giant linear nonreciprocity in a compact acoustic circulator
,”
Science
343
(
6170
),
516
519
(
2014
).
14.
T.
Kariyado
and
Y.
Hatsugai
, “
Manipulation of Dirac cones in mechanical graphene
,”
Sci. Rep.
5
,
18107
(
2015
).
15.
K. S.
Novoselov
,
A. K.
Geim
,
S.
Morozov
,
D.
Jiang
,
M.
Katsnelson
,
I.
Grigorieva
,
S.
Dubonos
, and,
A. A.
Firsov
, “
Two-dimensional gas of massless Dirac fermions in graphene
,”
Nature
438
(
7065
),
197
(
2005
).
16.
X.
Zhang
, “
Observing Zitterbewegung for photons near the direct point of a two-dimensional photonic crystal
,”
Phys. Rev. Lett.
100
(
11
),
113903
(
2008
).
17.
X.
Zhang
and
Z.
Liu
, “
Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals
,”
Phys. Rev. Lett.
101
(
26
),
264303
(
2008
).
18.
M.
Diem
,
T.
Koschny
, and
C.
Soukoulis
, “
Transmission in the vicinity of the Dirac point in hexagonal photonic crystals
,”
Phys. B (Amsterdam, Neth.)
405
(
14
),
2990
2995
(
2010
).
19.
T.
Ochiai
and
M.
Onoda
, “
Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states
,”
Phys. Rev. B
80
(
15
),
155103
(
2009
).
20.
J.
Mei
,
Y.
Wu
,
C. T.
Chan
, and
Z.-Q.
Zhang
, “
First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals
,”
Phys. Rev. B
86
(
3
),
035141
(
2012
).
21.
Y.
Li
,
Y.
Wu
, and
J.
Mei
, “
Double Dirac cones in phononic crystals
,”
Appl. Phys. Lett.
105
(
1
),
014107
(
2014
).
22.
Q.
Liang
,
Y.
Yan
, and
J.
Dong
, “
Zitterbewegung in the honeycomb photonic lattice
,”
Opt. Lett.
36
(
13
),
2513
2515
(
2011
).
23.
S. R.
Zandbergen
and
M. J.
de Dood
, “
Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene
,”
Phys. Rev. Lett.
104
(
4
),
043903
(
2010
).
24.
X.
Huang
,
Y.
Lai
,
Z. H.
Hang
,
H.
Zheng
, and
C.
Chan
, “
Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials
,”
Nat. Mater.
10
(
8
),
582
(
2011
).
25.
F.
Liu
,
X.
Huang
, and
C.
Chan
, “
Dirac cones at k = 0 in acoustic crystals and zero refractive index acoustic materials
,”
Appl. Phys. Lett.
100
(
7
),
071911
(
2012
).
26.
S.
Raghu
and
F.
Haldane
, “
Analogs of quantum-Hall-effect edge states in photonic crystals
,”
Phys. Rev. A
78
(
3
),
033834
(
2008
).
27.
Y.
Poo
,
R.-x.
Wu
,
Z.
Lin
,
Y.
Yang
, and
C.
Chan
, “
Experimental realization of self-guiding unidirectional electromagnetic edge states
,”
Phys. Rev. Lett.
106
(
9
),
093903
(
2011
).
28.
R.
Sepkhanov
and
C.
Beenakker
, “
Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure
,”
Opt. Commun.
281
(
20
),
5267
5270
(
2008
).
29.
S.
Xu
,
H.
Xu
,
H.
Gao
,
Y.
Jiang
,
F.
Yu
,
J. D.
Joannopoulos
,
M.
Soljačić
,
H.
Chen
,
H.
Sun
, and
B.
Zhang
, “
Broadband surface-wave transformation cloak
,”
Proc. Natl. Acad. Sci. U.S.A.
112
(
25
),
7635
7638
(
2015
).
30.
C.
Kane
and
T.
Lubensky
, “
Topological boundary modes in isostatic lattices
,”
Nat. Phys.
10
(
1
),
39
(
2014
).
31.
J.
Paulose
,
B. G.-g.
Chen
, and
V.
Vitelli
, “
Topological modes bound to dislocations in mechanical metamaterials
,”
Nat. Phys.
11
(
2
),
153
(
2015
).
32.
D. Z.
Rocklin
,
B. G.-g.
Chen
,
M.
Falk
,
V.
Vitelli
, and
T.
Lubensky
, “
Mechanical Weyl modes in topological Maxwell lattices
,”
Phys. Rev. Lett.
116
(
13
),
135503
(
2016
).
33.
O.
Stenull
,
C.
Kane
, and
T.
Lubensky
, “
Topological phonons and Weyl lines in three dimensions
,”
Phys. Rev. Lett.
117
(
6
),
068001
(
2016
).
34.
O. R.
Bilal
,
R.
Süsstrunk
,
C.
Daraio
, and
S. D.
Huber
, “
Intrinsically polar elastic metamaterials
,”
Adv. Mater.
29
(
26
),
1700540
(
2017
).
35.
E.
Prodan
,
K.
Dobiszewski
,
A.
Kanwal
,
J.
Palmieri
, and
C.
Prodan
, “
Dynamical Majorana edge modes in a broad class of topological mechanical systems
,”
Nat. Commun.
8
,
14587
(
2017
).
36.
A. B.
Khanikaev
,
R.
Fleury
,
S. H.
Mousavi
, and
A.
Alù
, “
Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice
,”
Nat. Commun.
6
,
8260
(
2015
).
37.
J.
Vila
,
R. K.
Pal
, and
M.
Ruzzene
, “
Observation of topological valley modes in an elastic hexagonal lattice
,”
Phys. Rev. B
96
(
13
),
134307
(
2017
).
38.
E.
Prodan
and
C.
Prodan
, “
Topological phonon modes and their role in dynamic instability of microtubules
,”
Phys. Rev. Lett.
103
(
24
),
248101
(
2009
).
39.
Y.-T.
Wang
,
P.-G.
Luan
, and
S.
Zhang
, “
Coriolis force induced topological order for classical mechanical vibrations
,”
New J. Phys.
17
(
7
),
073031
(
2015
).
40.
P.
Wang
,
L.
Lu
, and
K.
Bertoldi
, “
Topological phononic crystals with one-way elastic edge waves
,”
Phys. Rev. Lett.
115
(
10
),
104302
(
2015
).
41.
L. M.
Nash
,
D.
Kleckner
,
A.
Read
,
V.
Vitelli
,
A. M.
Turner
, and
W. T.
Irvine
, “
Topological mechanics of gyroscopic metamaterials
,”
Proc. Natl. Acad. Sci. U.S.A.
112
(
47
),
14495
14500
(
2015
).
42.
Z.
Yang
,
F.
Gao
,
X.
Shi
,
X.
Lin
,
Z.
Gao
,
Y.
Chong
, and
B.
Zhang
, “
Topological acoustics
,”
Phys. Rev. Lett.
114
(
11
),
114301
(
2015
).
43.
X.
Ni
,
C.
He
,
X.-C.
Sun
,
X.-p.
Liu
,
M.-H.
Lu
,
L.
Feng
, and
Y.-F.
Chen
, “
Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow
,”
New J. Phys.
17
(
5
),
053016
(
2015
).
44.
C.
Brendel
,
V.
Peano
,
O. J.
Painter
, and
F.
Marquardt
, “
Pseudomagnetic fields for sound at the nanoscale
,”
Proc. Natl. Acad. Sci. U.S.A.
17
,
E3390
E3395
(
2017
).
45.
R.
Chaunsali
,
E.
Kim
,
A.
Thakkar
,
P. G.
Kevrekidis
, and
J.
Yang
, “
Demonstrating an in situ topological band transition in cylindrical granular chains
,”
Phys. Rev. Lett.
119
(
2
),
024301
(
2017
).
46.
R.
Chaunsali
,
F.
Li
, and
J.
Yang
, “
Stress wave isolation by purely mechanical topological phononic crystals
,”
Sci. Rep.
6
,
30662
(
2016
).
47.
R.
Fleury
,
A. B.
Khanikaev
, and
A.
Alu
, “
Floquet topological insulators for sound
,”
Nat. Commun.
7
,
11744
(
2016
).
48.
N.
Swinteck
,
S.
Matsuo
,
K.
Runge
,
J.
Vasseur
,
P.
Lucas
, and
P. A.
Deymier
, “
Bulk elastic waves with unidirectional backscattering-immune topological states in a time-dependent superlattice
,”
J. Appl. Phys.
118
(
6
),
063103
(
2015
).
49.
L. D.
Landau
,
J.
Bell
,
M.
Kearsley
,
L.
Pitaevskii
,
E.
Lifshitz
, and
J.
Sykes
,
Electrodynamics of Continuous Media
(
Elsevier
,
2013
), Vol.
8
.
50.
C.
He
,
X.
Ni
,
H.
Ge
,
X.-C.
Sun
,
Y.-B.
Chen
,
M.-H.
Lu
,
X.-P.
Liu
, and
Y.-F.
Chen
, “
Acoustic topological insulator and robust one-way sound transport
,”
Nat. Phys.
12
(
12
),
1124
(
2016
).
51.
R.
Chaunsali
,
C.-W.
Chen
, and
J.
Yang
, “
Subwavelength and directional control of flexural waves in zone-folding induced topological plates
,”
Phys. Rev. B
97
(
5
),
054307
(
2018
).
52.
R. K.
Pal
and
M.
Ruzzene
, “
Edge waves in plates with resonators: An elastic analogue of the quantum valley Hall effect
,”
New J. Phys.
19
(
2
),
025001
(
2017
).
53.
C.
He
,
Z.
Li
,
X.
Ni
,
X.-C.
Sun
,
S.-Y.
Yu
,
M.-H.
Lu
,
X.-P.
Liu
, and
Y.-F.
Chen
, “
Topological phononic states of underwater sound based on coupled ring resonators
,”
Appl. Phys. Lett.
108
(
3
),
031904
(
2016
).
54.
S. D.
Huber
, “
Topological mechanics
,”
Nat. Phys.
12
(
7
),
621
(
2016
).
55.
R. K.
Pal
,
M.
Schaeffer
, and
M.
Ruzzene
, “
Helical edge states and topological phase transitions in phononic systems using bi-layered lattices
,”
J. Appl. Phys.
119
(
8
),
084305
(
2016
).
56.
J.
Ningyuan
,
C.
Owens
,
A.
Sommer
,
D.
Schuster
, and
J.
Simon
, “
Time-and site-resolved dynamics in a topological circuit
,”
Phys. Rev. X
5
(
2
),
021031
(
2015
).
57.
Y.
Hatsugai
, “
Chern number and edge states in the integer quantum Hall effect
,”
Phys. Rev. Lett.
71
(
22
),
3697
(
1993
).
58.
F.
Zhang
,
A. H.
MacDonald
, and
E. J.
Mele
, “
Valley Chern numbers and boundary modes in gapped bilayer graphene
,”
Proc. Natl. Acad. Sci. U.S.A.
110
(
26
),
10546
10551
(
2013
).
59.
A.
Rycerz
,
J.
Tworzydo
, and
C.
Beenakker
, “
Valley filter and valley valve in graphene
,”
Nat. Phys.
3
(
3
),
172
(
2007
).
60.
D.
Xiao
,
W.
Yao
, and
Q.
Niu
, “
Valley-contrasting physics in graphene: Magnetic moment and topological transport
,”
Phys. Rev. Lett.
99
(
23
),
236809
(
2007
).
61.
F.
Zhang
,
J.
Jung
,
G. A.
Fiete
,
Q.
Niu
, and
A. H.
MacDonald
, “
Spontaneous quantum Hall states in chirally stacked few-layer graphene systems
,”
Phys. Rev. Lett.
106
(
15
),
156801
(
2011
).
62.
W.
Yao
,
D.
Xiao
, and
Q.
Niu
, “
Valley-dependent optoelectronics from inversion symmetry breaking
,”
Phys. Rev. B
77
(
23
),
235406
(
2008
).
63.
K. F.
Mak
,
K. L.
McGill
,
J.
Park
, and
P. L.
McEuen
, “
The valley Hall effect in mos2 transistors
,”
Science
344
(
6191
),
1489
1492
(
2014
).
64.
R. V.
Gorbachev
,
J. C.
Song
,
G. L.
Yu
,
A. V.
Kretinin
,
F.
Withers
,
Y.
Cao
,
A.
Mishchenko
,
I. V.
Grigorieva
,
K. S.
Novoselov
,
L. S.
Levitov
, and
A. K.
Geim
, “
Detecting topological currents in graphene superlattices
,”
Science
346
(
6208
),
448
451
(
2014
).
65.
M.
Sui
,
G.
Chen
,
L.
Ma
,
W.-Y.
Shan
,
D.
Tian
,
K.
Watanabe
,
T.
Taniguchi
,
X.
Jin
,
W.
Yao
,
D.
Xiao
, and
Y.
Zhang
, “
Gate-tunable topological valley transport in bilayer graphene
,”
Nat. Phys.
11
(
12
),
1027
(
2015
).
66.
T.
Ma
and
G.
Shvets
, “
All-Si valley-Hall photonic topological insulator
,”
New J. Phys.
18
(
2
),
025012
(
2016
).
67.
J.-W.
Dong
,
X.-D.
Chen
,
H.
Zhu
,
Y.
Wang
, and
X.
Zhang
, “
Valley photonic crystals for control of spin and topology
,”
Nat. Mater.
16
(
3
),
298
(
2017
).
68.
M.
Xiao
,
W.-J.
Chen
,
W.-Y.
He
, and
C. T.
Chan
, “
Synthetic gauge flux and Weyl points in acoustic systems
,”
Nat. Phys.
11
(
11
),
920
(
2015
).
69.
Y.
Zhang
,
T.-T.
Tang
,
C.
Girit
,
Z.
Hao
,
M. C.
Martin
,
A.
Zettl
,
M. F.
Crommie
,
Y. R.
Shen
, and
F.
Wang
, “
Direct observation of a widely tunable bandgap in bilayer graphene
,”
Nature
459
(
7248
),
820
(
2009
).
70.
L.
Ju
,
Z.
Shi
,
N.
Nair
,
Y.
Lv
,
C.
Jin
,
J.
Velasco
, Jr.
,
C.
Ojeda-Aristizabal
,
H. A.
Bechtel
,
M. C.
Martin
,
A.
Zettl
,
J.
Analytis
, and
F.
Wang
, “
Topological valley transport at bilayer graphene domain walls
,”
Nature
520
(
7549
),
650
(
2015
).
71.
J.
Lu
,
C.
Qiu
,
M.
Ke
, and
Z.
Liu
, “
Valley vortex states in sonic crystals
,”
Phys. Rev. Lett.
116
(
9
),
093901
(
2016
).
72.
J.
Lu
,
C.
Qiu
,
L.
Ye
,
X.
Fan
,
M.
Ke
,
F.
Zhang
, and
Z.
Liu
, “
Observation of topological valley transport of sound in sonic crystals
,”
Nat. Phys.
13
(
4
),
369
(
2017
).
73.
K.
Qian
,
D. J.
Apigo
,
C.
Prodan
,
Y.
Barlas
, and
E.
Prodan
, “
Theory and experimental investigation of the quantum valley hall effect
,” arXiv:1803.08781 (
2018
).
74.
J.
Achenbach
,
Wave Propagation in Elastic Solids
(
Elsevier
,
North Holland
,
2012
), Vol.
16
.
77.
H.
Lamb
, “
On waves in an elastic plate
,”
Proc. R. Soc. Lond. A
93
(
648
),
114
128
(
1917
).
78.
B. I.
Halperin
, “
Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential
,”
Phys. Rev. B
25
(
4
),
2185
(
1982
).
79.
A. B.
Khanikaev
and
G.
Shvets
, “
Two-dimensional topological photonics
,”
Nat. Photon.
11
(
12
),
763
(
2017
).
80.
T.-W.
Liu
and
F.
Semperlotti
, “
Tunable acoustic valley–Hall edge states in reconfigurable phononic elastic waveguides
,”
Phys. Rev. Appl.
9
(
1
),
014001
(
2018
).
81.
See supplementary material at https://doi.org/10.1121/1.5114920 for detailed time responses of the system along the edges; along the triangular defect; and along the interface; and detailed time responses of the experimental setup along the edges; and the interface.
82.
H. C. P.
Adrian
, “
Electronic band structure in topological textures
,” Ph.D. thesis,
The Chinese University of Hong Kong
,
2011
.

Supplementary Material

You do not currently have access to this content.