Science - USA (2020-06-05)

The nonzero FCA in both metamaterials
indicates that they are indeed HOTIs, and
we have argued above that they should host
second-order topological modes at their corners.
Because we have not observed these expected
topological modes within the bulk bandgap,
we can estimate their spectral location by find-
ing the band in which the corner resonators
are most strongly excited. In theC 4 -symmetric
system, the corner resonators, around which
the second-order topological modes are ex-
pected to exist, are mainly excited in band 3,
which indicates that the corner modes lie in
this band. Moreover, this implies that we can
spectrally localize these modes by slightly
lowering the resonance frequency of the cor-
ner resonators. As illustrated in Fig. 3A, we
applied a small negative potential to the cor-
ner resonators using a capacitor connected to
ground, which decreases the electrical length
(and thus the resonance frequency) of the cor-
ner resonators. When the potential is applied,
we observe topological modes moving into the
bandgap between bands 2 and 3 and becom-
ing exponentially localized to the corner and
confined to one sublattice.
In theC 3 -symmetric system, the corner reso-
nators are only excited in band 2, which again
indicates that the energy of the corner modes
is too high and should be lowered to bring the
modes into the bandgap. We pull these modes
into the bandgap by similarly applying a small

negative potential to the corners, as illustrated

in Fig. 3B. The topological modes are observed

to spectrally localizewithin the bandgap and

spatially localize to the corners with confine-

ment on one sublattice. We also conducted a

similar experiment on a trivial insulator ( 30 )

and found that localized corner modes cannot

be isolated within the bulk bandgap, regard-

less of the applied potential strength.

The definition of the FCA can also be extended

beyond two dimensions to identifyd-th–order

topology in fully gappedd-dimensional insu-

lators. For example, the FCA for third-order

TCIs in three dimensions is

f¼

X

i;j

ðdrjsiÞmod 1 ð 3 Þ

wheredis corner-localized fractional mode

density,rjis hinge-localized fractional mode

density, andsiis surface-localized fractional

mode density. Because this indicator captures

fundamental topological features that are pro-

tected by spatial symmetries, we expect that it

can assist with the experimental identifica-

tion of materials with higher-order topology,

which could otherwise be misidentified by

only searching for in-gap corner modes. From

a practical perspective, focusing on bulk-

derived fractional mode density instead of

localized modes could simplify experimental

confirmations of novel TIs, which often use

ad hoc supplementary boundary elements (e.g.,

auxiliary resonators or loading capacitors) to

spectrally shift topological modes into the

bandgap ( 22 – 25 , 28 , 29 , 31 ).

REFERENCES AND NOTES

1. X.-L. Qi, S.-C. Zhang,Rev. Mod. Phys. 83 , 1057–1110 (2011).


2. M. Z. Hasan, C. L. Kane,Rev. Mod. Phys. 82 ,3045–3067 (2010).


3. A. P. Schnyder, S. Ryu, A. Furusaki, A. W. Ludwig,Phys. Rev. B
   78 , 195125 (2008).


4. R.-J. Slager, A. Mesaros, V. Juričić, J. Zaanen,Nat. Phys. 9 ,
   98 – 102 (2013).


5. Y. Ando, L. Fu,Annu.Rev.Condens.MatterPhys. 6 ,361–381 (2015).


6. W. A. Benalcazar, B. A. Bernevig, T. L. Hughes,Science 357 ,
   61 – 66 (2017).


7. Z. Song, Z. Fang, C. Fang,Phys. Rev. Lett. 119 , 246402 (2017).


8. F. Schindleret al.,Sci. Adv. 4 , eaat0346 (2018).


9. F. Schindleret al.,Nat. Phys. 14 , 918–924 (2018).


10. C. W. Peterson, W. A. Benalcazar, T. L. Hughes, G. Bahl,
    Nature 555 , 346–350 (2018).


11. S. Imhofet al.,Nat. Phys. 14 , 925–929 (2018).


12. H. Xue, Y. Yang, F. Gao, Y. Chong, B. Zhang,Nat. Mater. 18 ,
    108 – 112 (2019).


13. X. Ni, M. Weiner, A. Alù, A. B. Khanikaev,Nat. Mater. 18 ,
    113 – 120 (2019).


14. S. Mittalet al.,Nat. Photonics 13 , 692–696 (2019).


15. H. Xueet al.,Phys. Rev. Lett. 122 , 244301 (2019).


16. M. Weiner, X. Ni, M. Li, A. Alù, A. B. Khanikaev,Sci. Adv. 6 ,
    eaay4166 (2020).
    17.J. Nohet al.,Nat. Photonics 12 , 408–415 (2018).


17. A. El Hassanet al.,Nat. Photonics 13 , 697–700 (2019).


18. M. Serra-Garciaet al.,Nature 555 , 342–345 (2018).


19. X. Zhanget al.,Nat. Phys. 15 , 582–588 (2019).


20. Z.-K. Lin, H.-X. Wang, Z. Xiong, M.-H. Lu, J.-H. Jiang,
    Nonsymmorphic Quadrupole Topological Insulators in Sonic
    Crystals. arXiv:1903.05997 [cond-mat.str-el] (14 March 2019).


21. B. Y. Xieet al.,Phys. Rev. Lett. 122 , 233903 (2019).


22. X. D. Chenet al.,Phys. Rev. Lett. 122 , 233902 (2019).


23. W. A. Benalcazar, T. Li, T. L. Hughes,Phys. Rev. B 99 , 245151
    (2019).


24. J. Ahn, S. Park, B.-J. Yang,Phys. Rev. X 9 , 021013 (2019).

Petersonet al.,Science 368 , 1114–1118 (2020) 5 June 2020 4of5

Fig. 3. Pulling topological modes into the gap.(A) The schematic on the left shows where a small, negative on-site potential is applied (white circles). The
measured DOS has modes within the bulk bandgap, which are spatially localized to the corners and confined to one sublattice. (B) Same as (A) but for the
C 3 -symmetric insulator.

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