spectral information shown in Fig. 2A. Evalu-
ating the relative weight and phase at different
Wyckoff positions inside the unit cell allowed
us to extract the symmetry of the Bloch wave
functions (figs. S4 and S5) ( 35 ). In Fig. 2A, we
label the high-symmetry points according to the
irreducible representation of the respective little
group ( 37 ). The names and example orbitals of
the representationsare shown in Fig. 2B.
The measured symmetries confirm the ex-
pectations from the finite-element simulations.
We found the following decomposition ( 37 ):Bands 1 and 2:ðA 1 Þ 1 b⊕ðA 2 Þ 2 c⊖ðB 2 Þ 1 a ð 1 ÞBands 3 and 4:ðB 2 Þ 1 a⊕ðA 1 Þ 2 c⊖ðA 1 Þ 1 b ð 2 ÞThis establishes that the lowest two sets of
bands are fragile according to experimental
data alone. The bands 1 and 2 have the miss-
ing EBR induced from Wyckoff position 1a,
which does not host an acoustic cavity. In thiscase, fragility has a spectral consequence in
the form of spectral flow. We quickly review
how to establish this. Details can be found
in the companion paper ( 34 ).
The simplest approach is given by a real-
space picture ( 34 ). One can characterize all
states below the spectral gap of a finite sam-
ple by their transformation properties under
theC 2 symmetry around the central 1aposi-
tion. This allows us to write a real-space index
(RSI) ( 34 )Periet al.,Science 367 , 797–800 (2020) 14 February 2020 2of4
real spacereal spacemomentum spacecompareABCDEFHinduceinducesolvemeasure560 mmG40 mm
hopping multiplication factor1-1 0spectrumbulkbulkFig. 1. Atomic limit and topological bands.(A) Localized orbitals at the 2c
maximal Wyckoff positions in the unit cell of ap 4 mm-symmetric system.
(Top)A 1 orbitals. (Bottom)A 2 orbitals (Fig. 2B and table S3) ( 35 ). (B) Schematic
representation of the wallpaper groupp 4 mm. The locations and labels of
the maximal Wyckoff positions are in red. Dotted black lines indicate the relevant
mirror planes with respective labels, and solid lines show the action ofC 4 and
C 2 symmetry operators. (C) Sketch of the bands at high-symmetry points
induced by the localized orbitals of (A) ( 10 , 42 , 43 ). The drawings show example
orbitals that transform according to the realized irreducible representations.
(D) Labels and locations of the high-symmetry points in the Brillouin zone for the
p 4 mmwallpaper group. In parentheses are the little group realized at each
high-symmetry point. (E) Bands obtained from finite-element simulations of our
acoustic crystal. The irreducible representations at high-symmetry points are
represented by example orbitals. (F) (Top) A rendering of the air structure of the
acoustic crystal unit cell withp 4 mmsymmetry. The labels of the maximal
Wyckoff positions are in red. (Bottom) A lattice representation of the acoustic
structure. (G) Photo of the experimental sample, with the soft cut indicated
with the yellow dashed line. (Inset, top) A detail of the obstructions realizing the
cut. (Inset, bottom) A zoom-in of the unit cell. (H) Schematic of the flow
induced between fragile bands under twisted boundary conditions.Fig. 2. Irreducible representations at high-symmetry
points.(A) Measured spectrum of the acoustic crystal
along high-symmetry lines. The fit to the local maxima
has been overlaid at each point in momentum space,
and the vertical error bars are the full width at half
maxima of the fitted Lorentzians. Labels indicate the
irreducible representations of the little groupsGK
realized at high-symmetry pointsKaccording to the
names in (B) (GG=GM≅C 4 vandGX≅C 2 v). (Right) The
integrated density of states. The frequency range of
the bulk bands is shaded in gray. (B) Tables for the
irreducible representations ofC 4 vand its relevant
subgroups,C 2 vandC 2. The left column provides the
standard names according to ( 44 ), the middle column
provides the labels we gave the high-symmetry points
in the Brillouin zone (for example,K 1 →G 1 at the
Gpoint), and the right column depicts an example
orbital in the respective irreducible representation.
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