d=m(B)–m(A) (3)which counts the imbalance betweenC 2 -odd
andC 2 -even states (Fig. 2B). One can now
smoothly disconnect the sample in aC 2 -
symmetric way (Fig. 1G) and reconnect it
with the opposite hopping sign. The initial
and final dynamics are related by a gauge
transformation, in which we multiply all de-
greesoffreedominonehalfofthesampleby
minus one. However, this gauge transforma-
tion turnsC 2 -even intoC 2 -odd states. Hence,
we expectdstates flowing through the gap dur-
ing this cutting and reconnecting procedure.
The remaining task is to bind this RSIdto
the decomposition in Eq. 1. This can be achieved
with an exhaustive analysis of all EBRs in a
given space group, akin to the matching exer-
cise to establish fragility ( 34 ). For our situa-
tion, we found ( 34 )
d= 1 (4)To implement this soft cut in our sample, we
inserted obstructions of growing sizes into
the channels along the yellow dashed line in
Fig. 1G. Without using higher-order resonances
in the connecting channels ( 38 ), we could only
perform half of the twisting cycle: from the
original sample (l= 1) to a total disconnect
(l=0),wherelis the hopping multiplication
factoracrossthecut.However,thespectrum
atland–l is related by a gauge transfor-
mation. Hence, the pathl=0→−1 does
not provide further information ( 35 ).
The measured spectral flow is shown in
Fig. 3, which depicts how a state is flowing
from the upper bulk bands to the lower bulk
bands in the course of the cutting procedure.
We obtained these results from the local den-
sity of states measured at the symmetry cen-
ter of the sample. This measurement gives a
clean and robust spectralsignature associated
tofragiletopology.TheoppositeflowofaC 2 -
odd state from the lower bands to upper ones
occurs forl< 0 as ensured by the symmetry
l→–l( 34 , 35 ).
Through the results presented here, we
achieved two goals. First, we established the
presence of fragile bands through full band
tomography. Second, we demonstrate that fra-
gile topology, a concept that challenges our
understanding of topology, can yield a clean
and simple experimental observable. In the
context of metamaterials design, such fine
control of well-localized states is an impor-
tant building block for mechanical logic and
other wave-control applications ( 39 ). More-
over, many classical topological metamateri-
als rely on crystalline symmetries ( 40 ). This
attributes fragility a more prominent role
than anticipated ( 18 , 32 , 35 ). Last, the ex-
pectation that fragility plays a role in the
reported strongly correlated superconductiv-
ity in twisted bilayer graphene ( 27 – 31 , 41 )
raises the natural question of how classical
nonlinearities are influenced by these intri-
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ACKNOWLEDGMENTS
We thank G. Blatter, T. Neupert, and V. Vitelli for insightful
discussions.Funding:S.D.H., V.P., M.S.-G., and P.E. acknowledge
support from the Swiss National Science Foundation, the Swiss
National Center of Competence in Research QSIT, and the European
Research Council under grant agreement 771503 (TopMechMat).
Z.-D.S. and B.A.B. are supported by the Department of Energy grant
DE-SC0016239, the National Science Foundation (NSF) EAGER
grant DMR1643312, Simons Investigator Grant 404513, Office of Naval
Research N00014-14-1-0330, NSF–Materials Research Science andPeriet al.,Science 367 , 797–800 (2020) 14 February 2020 3of4
Fig. 3. Spectral flow between fragile bands.Measured local density of state at the symmetry center
of the acoustic crystal for different valueslof the hopping multiplication factor across the cut. The overlaid
dots indicate the fit to the local maxima, and the vertical error bars correspond to the full width at half
maxima of the fitted Lorentzians. The frequency range of the bulk bands has been shaded in gray. The data
at eachlhave been individually normalized with respect to their maximum value.
RESEARCH | REPORT