Science - USA (2020-06-05)

such thatf
ð 4 Þ
1 —i.e., the FCA for band 1 in the
C 4 -symmetric metamaterial—is

fð 14 Þ¼rð 14 Þ 2 sð 14 Þ¼ 0 : 23  0 : 98

¼ 0 : 25 ≈

1

4

ð 2 Þ

A similar calculation can be carried out for
the other bulk bands, givingf
ð 4 Þ
2 ¼^0 :^55 ≈
1
2
for band 2 andfð 34 Þ¼ 0 : 19 ≈^14 for band 3. The
sum of the FCA over all the bulk bands is al-
ways an integer (here rounding givesfð 14 Þþ

f

ð 4 Þ

2 þf

ð 4 Þ

3 ¼^0 :99), andf

ð 4 Þ

2 is twicef

ð 4 Þ

1 ,as

expected.

Next, we move on to the mode density for

theC 3 -symmetric system,showninFig.2B.

For this material, we again find that the bulk

unit cells have integer mode density,mð 13 Þ≈ 2

(a twofold degenerate band) andmð 13 Þ≈1. As in

the previous system, the total mode density of

these bands in each sector is approximately

equal. The edge unit cells have a fractional

mode density ofsð 13 Þ≈^23 in band 1 andsð 23 Þ≈^13

in band 2. Here, the approximate fractions

are obtained by rounding to the nearest third

because this system isC 3 symmetric ( 24 ).

Notably, although this material does not have

a fractional mode density in the corner unit

cells, the FCA indicator is nonzero,fð 13 Þ¼

0 : 70 ≈^23 ,andfð 23 Þ¼ 0 : 30 ≈ 31. For comparison,

in the supplementary materials ( 30 )wealso

present experimental measurements of the

FCA for a trivial insulator and show that

fð^3 Þ;triv≈0 for all bands.

Petersonet al.,Science 368 , 1114–1118 (2020) 5 June 2020 3of5

Fig. 2. Spatial distribution of bands.(A) Measured mode density for theC 4 -symmetric insulator. The mode density for each band is shown separately. Each filled circle
represents a resonator, with the area of the circle corresponding to the measured mode density of that resonator in that band. The total mode density of each unit cell
is shown with black text, and the total mode density of each sector is shown in purple with white text. (B) Same as (A) but for theC 3 -symmetric insulator.

RESEARCH | REPORT