an EBR at the empty Wyckoff position, where
no atom exists, it forms an obstructed atom-
ic insulator band. The model, in the basis
ðpx;py;s 1 ;s 2 Þ,is( 27 )HðkÞ¼tzs 0
Eþ 2 t 1 cosðkxþkyÞþ2 t 1 cosðkxkyÞ
þtyszt 2 sinðkxÞþtysxt 2 sinðkyÞð 1 ÞE(E) is the onsite energy for the px,y(s1,2)
orbitals,t 1 band inverts atX,andt 2 guarantees
a full gap between the upper and lower two
bands. We introduceDHðkÞ( 27 )tobreaktwo
accidental symmetries,Mz(z→z)¼tzsy,
chiraltxs 0 .ThebandstructureofHðkÞþ
DHðkÞis plotted in Fig. 2B.
We construct a finite-size (30 × 30) TRS
Hamiltonian withC 4 rotation symmetry pre-
served at the coordinate origin on theasite
(Fig. 2C). The spectrum consists of 1798 oc-
cupied states, 4 degenerate partially occupied
levels at the Fermi level, and 1798 empty
levels; they form the representations 450A⊕
450 B⊕ 449 ð^1 E^2 EÞ,A⊕B⊕^1 E^2 E, and 449A⊕
449 B⊕ 450 ð^1 E^2 EÞ, respectively. The partially
occupied states are corner states, or the“fil-
ling anomaly”of fragile topology (Fig. 2C).
The gap between the four partially occupied
levels and the occupied or empty levels is
about 0: 3 = 0 :01, asDHðkÞbreaks the acci-
dental chiral symmetry. Every four states
forming the irrepsA⊕B⊕^1 E^2 Ecan be recom-
bined asj 1 i¼ðjAiþjBiþj^1 Eiþj^2 EiÞ=2,
j 2 i¼ðjAijBiij^1 Eiþij^2 EiÞ=2,j 3 i¼ðjAiþ
jBij^1 Ei j^2 EiÞ=2, andj 4 i¼ðjAijBiþ
ij^1 Eiij^2 EiÞ=2, which transform to each other
under theC 4 rotation and have Wannier cen-
ters away from theC 4 center: We hence move the
occupied and empty states, both of which form
the representation 449A⊕ 449 B⊕ 449 ð^1 E^2 EÞ,
away from theC 4 center. We are left with two
occupied states,A⊕B, and two empty states,
ð^1 E^2 EÞ,attheC 4 center. These four states form
a level crossing under TBC evolution.
We divide (Fig. 2C) the system into four
parts (m¼I, II, III, IV), which transform into
each other underC 4. We introduce the TBC
by multiplying the hoppings between differ-
ent parts by specific factors such that the
twisted and original Hamiltonians are equiv-
alent up to a gauge transformation. Specifi-
cally, the multiplication factors on hoppings
frommth part toðmþ 1 Þth part, frommth part
toðmþ 2 Þth part, frommth part toðm 1 Þth
part arei; 1 ;i. The twisted and untwisted
HamiltoniansH^ðiÞ;H^ð 1 Þsatisfyhm;ajH^ðiÞjn;bi≡ðiÞnmhm;ajH^ð 1 Þjn;bið 2 Þjm;aiis theath orbital in themth part, andHðlÞ
is the Hamiltonian with multiplierl.Weintro-
duce the twisted basisV^jm;ai¼ðiÞm^1 jm;ai.
The elements ofH^ðiÞon the twisted basis
equal those ofH^ð 1 Þon the untwisted basis:
hm;ajV^†^
HðiÞV^jn;bi¼hm;ajH^ð 1 Þjn;bi.C 4 trans-
forms themth part into theðmþ 1 Þth part: The
twisting phases ofjm;aiandC^ 4 jm;aiunder
V^ areðiÞðm^1 ÞandðiÞm,implyingV^C^ 4 ¼
i^C 4 V^.Ifjyiis an eigenstate ofH^ð 1 ÞwithC 4
eigenvaluex,thenV^jyiwill be an eigenstateofH^ðiÞ¼V^H^ð 1 ÞV^†
of equal energy but dif-
ferentC 4 eigenvalueix. The irrepsA,B,^1 E,^2 E
become^2 E,^1 E,A,Bunder the gauge trans-
formation (Table 1). Therefore, two of the ir-
repsA⊕Bin the occupied states interchange
with two of the irreps^1 E⊕^2 Ein the empty states
after the gauge transformation; all other irreps,Songet al.,Science 367 , 794–797 (2020) 14 February 2020 2of4Fig. 1. EBRs of wallpaper groupp4 without SOC with TRS.[See BCS ( 1 , 2 , 28 )]. The square represents the unit
cell.að 0 ; 0 Þ,b^12 ;^12
,c 0 ; 21
; 21 ; 0
are maximal Wyckoff positions. The yellow, red and blue, and green and grayorbitals represent thes,px;y,anddx (^2) y 2 orbitals, respectively.
Fig. 2. Spectral flow of fragile phase under TBCs.(A) Fragile phase model (wallpaper groupp4withTRS).The
yellow, green, and red and blue orbitals are the two s and the px;yorbitals. The gray parallelogram is the unit cell, and
black lines are the hoppings. (B) Band structure of the fragile phase. (C)TheC 4 -symmetric TBCs of a finite size system.
Black dots are the atoms; bonds are hoppings; four yellow circles are corner states of the fragile state. Four shaded regions
(m¼I, II, III, IV) transform to each other underC 4 action. Hoppings from themth part to theðmþ 1 Þth part (red bonds),
from themth part to theðmþ 2 Þth part (green bonds), and from themth part to theðm 1 Þth part (red bonds)
are multiplied by a complexl/real Reðl^2 Þ/complexl.(D)TheC 2 and TRS symmetricTBCs. The two shaded regions
(m¼I, II) transform to each other underC 2 rotation. The hoppings between I,II (red bonds) are multiplied by a
real numberl.(E) Spectral flow underC 4 -symmetric TBC. (F) Spectral flow underC 2 and TRS symmetric TBCs.
RESEARCH | REPORT