Science 14Feb2020

(449A⊕ 449 B⊕ 449 ð^1 EÞ⊕ 449 ð^2 EÞ), remain un-
changed. We generalize theC 4 symmetric TBC:

hm;ajH^ðlÞjn;bi¼
hm;ajH^ð 1 Þjn;bi; n¼m
lhm;ajH^ð 1 Þjn;bi; n¼mþ1mod4
lhm;ajH^ð 1 Þjn;bi; n¼m1mod4
Reðl^2 Þhm;ajH^ð 1 Þjn;bi; n¼mþ2mod4

0

B

B

B

B

@

ð 3 Þ

Reðl^2 Þis the real part of the complexl^2 .The
factor between themth andðmþ 2 Þth part is
real owing toC 2 ( 27 ). Equation 2 is thel¼i
case of Eq. 3. Under continuous tuning ofl
from 1 toi, two occupied irrepsA⊕Binter-
change with two empty irreps^1 E⊕^2 E. Their
level crossings are protected byC 4 symmetry
(Fig. 2E) [see ( 27 ) for otherC 4 paths].
Now we considerC 4 -breaking butC 2 -andTRS-
preserving TBCs. Divide the system into two parts
(I,II), transforming into each other underC 2 (Fig.
2D), and multiply all hoppings between orbitals
in part I and II by a reall. The gauge trans-
formation relating the twisted and untwisted
Hamiltonians anticommutes withC 2 :fV^;^C 2 g¼
0.V^transforms between eigenstates ofH^ð 1 Þ;
H^ð 1 Þwith equal energy but oppositeC 2 eigen-
value. Under a continuous tuning oflfrom 1 to
1, the two final occupied (empty) states must
have theC 2 eigenvalue1(1). This inconsistency
impliesC 2 -protected gap closing, as shown in Fig.
2F. The unitary transformation relatingHð 1 Þ
toHð 1 Þalso mapsHðlÞtoHðlÞ,andthegap
must close aslchanges from 1 to 0. In ( 27 ), we
generalize the TBCs to all the 2D PGs. The
gapless states under TBCs are the experimen-
tal consequences of the fragile states.
We introduce the RSI as an exhaustive de-
scription of the local states, pinned at theC 4
center, that undergo gap closing under TBCs.
The Wannier centers of occupied states of a
Hamiltonian can adiabatically move if their
displacements preserve symmetry. Orbitals
away from a symmetry center can move on it
and form an induced representation of the
site-symmetry group at the center. Conversely,
orbitals at a symmetry center can move away
from it symmetrically if and only if they form a
representation induced from the site-symmetry
groups away from the center. The RSIs are
( 27 ) linear invariant—upon such induction
processes—functions of irrep multiplicities.

For the PG 4 with TRS, we assume a linear-

form RSI of the occupied levelsd¼c 1 mðAÞþ

c 2 mðBÞþc 3 mð^1 E^2 EÞ. The induced represen-

tation at theC 4 center from four states atC 4 -

related positions away from the center is

A⊕B⊕^1 E^2 E( 27 ). After the induction pro-

cess, the irrep multiplicities at theC 4 center

change asmðAÞ→mðAÞþ1,mðBÞ→mðBÞþ

1,mð^1 E^2 EÞ→mð^1 E^2 EÞþ1. The two linear

combinations of irrep multiplicities that re-

main invariant are

d 1 ¼mð^1 E^2 EÞmðAÞ;

d 2 ¼mðBÞmðAÞð 4 Þ

In our model, the occupied states that can be

moved away from theC 4 center form the rep-

resentation 449A⊕ 449 B⊕ 449 ð^1 E^2 EÞand have

vanishing RSIs. The states pinned at theC 4

center formA⊕Bwith RSIsd 1 ¼1,d¼0. If

an RSI is nonzero, spectral flow exists upon a

particular TBC ( 27 ). We calculate all the RSIs

in all 2D PGs with and without SOC or TRS

[table S4 ( 27 )]. The groups formed by RSIs are

shown in Table 2. PG 4 with TRS has two

integer-valued RSIs: The RSI group isℤ^2 .Most

RSIs areℤ-type; some groups with SOC and

TRS haveℤ 2 -type RSIs, the parities of the

number of occupied Kramer pairs.

For theC 4 -symmetric TBC ( 3 ), the occupied

irrep multiplicitiesm′atl¼iare determined

by the multiplicitiesmatl¼1asm′ðAÞ¼

mð^1 EÞ,m′ðBÞ¼mð^2 EÞ,m′ð^1 EÞ¼mðBÞ,and

m′ð^2 EÞ¼mðAÞ. The changes of irreps in the

evolutionl¼ 1 →iareDmðAÞ¼m′ðAÞ

mðAÞ¼mð^1 EÞmðAÞ¼d 1 , DmðBÞ¼d 1 

d 2 ,Dmð^1 EÞ¼d 2 d 1 ,andDmð^2 EÞ¼d 1.

Therefore, there will bejd 1 jcrossings formed

byAand^2 Eandjd 2 d 1 jcrossings formed by

Band^1 E. This and the similar analysis forC 2

and TRS-symmetric TBCs are given in Tables 3

and 4 and expanded in ( 27 ). Our model (d 1 ¼

1andd 2 ¼0)hastwolevelcrossingspro-

tected byC 2 in the processl¼ 1 →1.

The RSI can be calculated either from the

momentum space irreps of the band structure

or from symmetry-center PG-respecting dis-

ordered configurations. In ( 27 ), we develop a

general framework to calculate the RSIs from

momentum space irreps and obtain the expres-

sions of RSIs in all wallpaper groups [table S5

( 27 )]. Here we give the expressions for RSIs of

wallpaper groupp4.p4 has two inequivalent

C 4 Wyckoff positions,aandb,andoneC 2

Wyckoff position,c(Fig. 1). PG 4 has two RSIs,

d 1 andd 2 (Eq. 4), and PG 2 has a single RSI,

d 1 ¼mðBÞmðAÞ[table S2 ( 27 )]. The band

structure expressions are

da 1 ¼mðG 1 Þ

mðG 2 Þ

2

mðG 3 G 4 Þþ

mðM 2 Þ

2

þmðM 3 M 4 Þþ

mðX 2 Þ

2

ð 5 Þ

da 2 ¼mðG 1 ÞmðG 3 G 4 ÞþmðM 2 Þþ

mðM 3 M 4 Þð 6 Þ

db 1 ¼

1

2

mðG 2 ÞþmðG 3 G 4 Þ

1

2

mðM 2 Þ

1

2

mðX 2 Þð 7 Þ

db 2 ¼mðG 2 ÞþmðG 3 G 4 ÞmðM 2 Þ

mðM 3 M 4 Þð 8 Þ

dc 1 ¼mðG 3 G 4 ÞmðM 3 M 4 Þð 9 Þ

One can immediately verify that the band

structure shown in Fig. 2B has the RSIs

da 1 ¼1,da 2 ¼0attheaposition, which

are the same as the results calculated from

the disordered configuration.

We find that the RSIs fully describe eigen-

value band topology: EFP is diagnosed by in-

equalities or mod-equations of RSIs ( 15 ), and

stable topology is diagnosed by fractional RSIs.

We prove this in all the wallpaper groups in

( 27 ). For the wallpaper groupp4, the stable

topology is diagnosed by fractionalda 1 anddb 1 ,

which imply topological semimetal phase with

Dirac nodes at general momenta ( 30 ). The

fragile topology, by contrast, is diagnosed by

the inequality

N<maxð 2 jda 1 jþda 2 ; 2 da 1  3 da 2 Þþ

maxð 2 jdb 1 jþdb 2 ; 2 db 1  3 db 2 Þþjdc 1 jð 10 Þ

whereNis the number of bands. When this

inequality is fullfilled, the RSIs and band num-

ber are not consistent with any Wannierizable

Songet al.,Science 367 , 794–797 (2020) 14 February 2020 3of4

Table 3.

C 4 :l 1 →i

D.............................................................................................mðAÞ d 1

D.............................................................................................mð^1 EÞ d 2 d 1

D.............................................................................................mðBÞ d 1 d 2

D.............................................................................................mð^2 EÞd 1

Table 4.

C 2 ∈R 1 →− 1

D.............................................................................................mðAÞ 2 d 1 d 2

D.............................................................................................mðBÞ 2 d 1 þd 2

Table 2. The RSI groups of 2D PGs.

SOC TRS 2 m 2 mm 44 mm 33 m 66 mm

.....................................................................................................................................................................................................................✗✗ZZZZ^3 Z^2 Z^2 ZZ^5 Z^3

.....................................................................................................................................................................................................................✗ ✓ ZZZZ^2 ZZZZ^3 Z^3

.....................................................................................................................................................................................................................✓ ✗ ZZZ^1 Z^3 ZZ^2 ZZ^5 Z^2

.....................................................................................................................................................................................................................✓✓Z^2 Z^2 Z^2 Z^2 ZZ^2 ZZZZ^2 Z^2 Z^2 Z^2

RESEARCH | REPORT