We evaluated topological protection for edge and higher-order corner states in topological diffusion systems based on the breathing kagome lattice. In the kagome lattice, the corner states appear at the corner boundary where all three Wannier centers in nontrivial unit cells are located. The three Wannier centers in a unit cell can be placed on the obtuse- and acute-corner boundaries utilizing the armchair boundary, generating topological acute- and obtuse-corner states. For another representative zigzag boundary, only the acute-corner unit cell has three Wannier centers located at the boundary; hence, only the acute-corner state appears. Our band analysis and numerical studies show that the topologically protected decay behavior for armchair boundaries is as robust as that for zigzag boundaries, unlike wave phenomena with space and time periodicities. Our findings can guide the flexible design of topological diffusion applications such as heat localization and recovery systems.
REFERENCES
1.
C. H.
Lee
,
S.
Imhof
,
C.
Berger
,
F.
Bayer
,
J.
Brehm
,
L. W.
Molenkamp
,
T.
Kiessling
, and
R.
Thomale
, “
Topolectrical circuits
,” Commun. Phys.
1
, 39
(2018
).2.
Y.
Li
,
Y.
Sun
,
W.
Zhu
,
Z.
Guo
,
J.
Jiang
,
T.
Kariyado
,
H.
Chen
, and
X.
Hu
, “
Topological LC-circuits based on microstrips and observation of electromagnetic modes with orbital angular momentum
,” Nat. Commun.
9
, 4598
(2018
).3.
L.
Lu
,
J. D.
Joannopoulos
, and
M.
Soljačić
, “
Topological photonics
,” Nat. Photonics
8
, 821
–829
(2014
).4.
T.
Ozawa
,
H. M.
Price
,
A.
Amo
,
N.
Goldman
,
M.
Hafezi
,
L.
Lu
,
M. C.
Rechtsman
,
D.
Schuster
,
J.
Simon
,
O.
Zilberberg
, and
I.
Carusotto
, “
Topological photonics
,” Rev. Mod. Phys.
91
, 015006
(2019
).5.
Z.
Yang
,
F.
Gao
,
X.
Shi
,
X.
Lin
,
Z.
Gao
,
Y.
Chong
, and
B.
Zhang
, “
Topological acoustics
,” Phys. Rev. Lett.
114
, 114301
(2015
).6.
X.
Zhang
,
M.
Xiao
,
Y.
Cheng
,
M.-H.
Lu
, and
J.
Christensen
, “
Topological sound
,” Commun. Phys.
1
, 97
(2018
).7.
J.
Cha
,
K. W.
Kim
, and
C.
Daraio
, “
Experimental realization of on-chip topological nanoelectromechanical metamaterials
,” Nature
564
, 229
–233
(2018
).8.
R. J.
Davis
,
D. J.
Bisharat
, and
D. F.
Sievenpiper
, “
Classical-to-topological transmission line couplers
,” Appl. Phys. Lett.
118
, 131102
(2021
).9.
M.
Miniaci
and
R. K.
Pal
, “
Design of topological elastic waveguides
,” J. Appl. Phys.
130
, 141101
(2021
).10.
K.
Funayama
,
K.
Yatsugi
,
A.
Miura
, and
H.
Iizuka
, “
Control of coupling between micromechanical topological waveguides
,” Int. J. Mech. Sci.
236
, 107755
(2022
).11.
S. D.
Huber
, “
Topological mechanics
,” Nat. Phys.
12
, 621
–623
(2016
).12.
G.
Ma
,
M.
Xiao
, and
C. T.
Chan
, “
Topological phases in acoustic and mechanical systems
,” Nat. Rev. Phys.
1
, 281
–294
(2019
).13.
Y.
Li
,
Y.-G.
Peng
,
L.
Han
,
M.-A.
Miri
,
W.
Li
,
M.
Xiao
,
X.-F.
Zhu
,
J.
Zhao
,
A.
Alù
,
S.
Fan
, and
C.-W.
Qiu
, “
Anti–parity-time symmetry in diffusive systems
,” Science
364
, 170
–173
(2019
).14.
H.
Hu
,
S.
Han
,
Y.
Yang
,
D.
Liu
,
H.
Xue
,
G. G.
Liu
,
Z.
Cheng
,
Q. J.
Wang
,
S.
Zhang
,
B.
Zhang
, and
Y.
Luo
, “
Observation of topological edge states in thermal diffusion
,” Adv. Mater.
34
, 2202257
(2022
).15.
H.
Wu
,
H.
Hu
,
X.
Wang
,
Z.
Xu
,
B.
Zhang
,
Q. J.
Wang
,
Y.
Zheng
,
J.
Zhang
,
T. J.
Cui
, and
Y.
Luo
, “
Higher-order topological states in thermal diffusion
,” Adv. Mater.
35
, 2210825
(2023
).16.
T.
Yoshida
and
Y.
Hatsugai
, “
Bulk-edge correspondence of classical diffusion phenomena
,” Sci. Rep.
11
, 888
(2021
).17.
M.
Qi
,
D.
Wang
,
P.-C.
Cao
,
X.-F.
Zhu
,
C.-W.
Qiu
,
H.
Chen
, and
Y.
Li
, “
Geometric phase and localized heat diffusion
,” Adv. Mater.
34
, 2202241
(2022
).18.
T.
Fukui
,
T.
Yoshida
, and
Y.
Hatsugai
, “
Higher-order topological heat conduction on a lattice for detection of corner states
,” Phys. Rev. E
108
, 024112
(2023
).19.
K.
Funayama
,
J.
Hirotani
,
A.
Miura
, and
H.
Tanaka
, “
Robustness of quantum spin hall effect-inspired edge modes depending on C6 symmetry in topological diffusion systems
,” Appl. Phys. Lett.
123
, 223104
(2023
).20.
K.
Funayama
,
J.
Hirotani
,
A.
Miura
, and
H.
Tanaka
, “
Selectable diffusion direction with topologically protected edge modes
,” Commun. Phys.
6
, 364
(2023
).21.
G.
Xu
,
Y.
Yang
,
X.
Zhou
,
H.
Chen
,
A.
Alù
, and
C.-W.
Qiu
, “
Diffusive topological transport in spatiotemporal thermal lattices
,” Nat. Phys.
18
, 450
–456
(2022
).22.
G.
Xu
,
X.
Zhou
,
S.
Yang
,
J.
Wu
, and
C.-W.
Qiu
, “
Observation of bulk quadrupole in topological heat transport
,” Nat. Commun.
14
, 3252
(2023
).23.
B.
Orazbayev
and
R.
Fleury
, “
Quantitative robustness analysis of topological edge modes in C6 and valley-hall metamaterial waveguides
,” Nanophotonics
8
, 1433
–1441
(2019
).24.
Y.
Jin
,
W.
Wang
,
Z.
Wen
,
D.
Torrent
, and
B.
Djafari-Rouhani
, “
Topological states in twisted pillared phononic plates
,” Extreme Mech. Lett.
39
, 100777
(2020
).25.
H.-X.
Wang
,
G.-Y.
Guo
, and
J.-H.
Jiang
, “
Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology
,” New J. Phys.
21
, 093029
(2019
).26.
M.
Blanco de Paz
,
C.
Devescovi
,
G.
Giedke
,
J. J.
Saenz
,
M. G.
Vergniory
,
B.
Bradlyn
,
D.
Bercioux
, and
A.
García-Etxarri
, “
Tutorial: Computing topological invariants in 2D photonic crystals
,” Adv. Quantum Technol.
3
, 1900117
(2020
).27.
H.
Xue
,
Y.
Yang
,
F.
Gao
,
Y.
Chong
, and
B.
Zhang
, “
Acoustic higher-order topological insulator on a kagome lattice
,” Nat. Mater.
18
, 108
–112
(2019
).28.
M.
Li
,
D.
Zhirihin
,
M.
Gorlach
,
X.
Ni
,
D.
Filonov
,
A.
Slobozhanyuk
,
A.
Alù
, and
A. B.
Khanikaev
, “
Higher-order topological states in photonic kagome crystals with long-range interactions
,” Nat. Photonics
14
, 89
–94
(2020
).29.
Z.
Zheng
,
J.
Yin
,
J.
Wen
, and
D.
Yu
, “
Higher-order topological states in locally resonant elastic metamaterials
,” Appl. Phys. Lett.
120
, 144101
(2022
).30.
H.
Fan
,
B.
Xia
,
L.
Tong
,
S.
Zheng
, and
D.
Yu
, “
Elastic higher-order topological insulator with topologically protected corner states
,” Phys. Rev. Lett.
122
, 204301
(2019
).31.
J.
Ma
,
X.
Xi
, and
X.
Sun
, “
Experimental demonstration of dual-band nano-electromechanical valley-Hall topological metamaterials
,” Adv. Mater.
33
, 2006521
(2021
).32.
J.
Noh
,
S.
Huang
,
K. P.
Chen
, and
M. C.
Rechtsman
, “
Observation of photonic topological valley Hall edge states
,” Phys. Rev. Lett.
120
, 063902
(2018
).33.
H.
Danawe
,
H.
Li
,
H. A.
Ba'ba'a
, and
S.
Tol
, “
Existence of corner modes in elastic twisted kagome lattices
,” Phys. Rev. B
104
, L241107
(2021
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
