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# Topological crystalline insulator: from symmetry indicators to material discovery

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2018/Dec./6

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### 1. Topology in condensed matter physics

Basic concepts and properties of topological band structure Example: Bi2Se3

### 2. Topological crystalline insulator

Recent prediction: Ca2As family

### 3. Type-II nodal line semimetal

Recent prediction: Mg3Bi2

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slides number

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### Math => real space

Gauss-Bonnet Theorem:

genus g = 0 g = 1 g = 1

### Phys => momentum space

A B

B A

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B A

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http://pb.ps-taiwan.org/catalog/ins.php?index_m1_id=5&index_id=235

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### Topology in condensed matter physics

A-orbital B-orbital (1)

Brillouin zone

wavefunction is smoothly

Brillouin zone (2)

wavefunction is NOT smoothly A-orbital B-orbital

band inversion

band inversion (Se-orbital)

(Bi-orbital)

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Obey chemical bond order

Violate chemical bond order

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### Topology in condensed matter physics

Gauss-Bonnet Theorem:

TKNN theory: Topological invariant number

### Math => real spacePhys => momentum space

Berry curvature

wavefunction is smoothly : n = 0 =>

wavefunction is NOT smoothly : n = integer =>

Topological trivial (normal insulator)

Topological nontrivial (topological insulator)

Bloch wavefun

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Γ 𝑋

𝑋 𝑀 𝑅

𝑈 𝑈

𝑍

+Γ(+) +

−−

−X(+)

− ++

−X(+)

− ++

−M(+)

− ++

+Z(+) +

−−

−U(+)

− ++

−U(+)

− ++

−R(+)

− ++

EF

Γ 𝑋

𝑋 𝑀 𝑅

𝑈 𝑈

𝑍

+Γ(−)

− +−

−X(+)

− ++

−X(+)

− ++

−M(+)

− ++

+Z(+) +

−−

−U(+)

− ++

−U(+)

− ++

−R(+)

− ++

EF Band inversion at Γ

Normal insulator (Z2=0)

Topological insulator (Z2=1)

L. Fu, PRB 76, 045302 (2007)

### Calculating invariant number

𝛿Γ𝛿𝑋𝛿𝑀𝛿𝑌𝛿𝑍𝛿𝑈𝛿𝑇𝛿𝑅 = (−1)𝑍2

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### Topology in condensed matter physics (bulk-edge correspondence)

Surface/Interface

### p

gapless surface state

### s

inversion point (gapless point) electron move

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)

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Z2=0 Z2=1

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### ARPESTheory

Surface: gapless surface states spin-momentum locked Bulk: insulating gap

topological Z2 invariant

odd/even number surface states

CB SS

CB SS

8

CB band

inversion

Y. Xia et al. Nature Physics 5, 398 (2009) D. Hsieh et al. Nature 460, 1101 (2009)

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### Topological crystalline insulator

Γ 𝑋

𝑋 𝑀 𝑅

𝑈 𝑈

𝑍

+Γ(+) +

−−

−X(+)

− ++

−X(+)

− ++

−M(+)

− ++

+Z(+) +

−−

−U(+)

− ++

−U(+)

− ++

−R(+)

− ++

EF

Γ 𝑋

𝑋 𝑀 𝑅

𝑈 𝑈

𝑍

+Γ(+) +

−−

−X(−) +

−+

−X(−) +

−+

−M(+)

− ++

+Z(+) +

−−

−U(+) +

−+

−U(+) +

−+

−R(+)

− ++

EF Band inversion at 𝑋and 𝑈

Normal insulator (Z2=0)

??? (Z2=0)

L. Fu, PRB 76, 045302 (2007)

L. Fu, PRL 106, 106802 (2011)

### Normal band insulator?

𝛿Γ𝛿𝑋𝛿𝑀𝛿𝑌𝛿𝑍𝛿𝑈𝛿𝑇𝛿𝑅 = (−1)𝑍2

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### Topological crystalline insulator

Band inversion

odd

even

Topological insulator

no

yes

Normal band insulator

Topological crystalline

Insulator (TCI) (Z2=1)

(Z2=0)

(Z2=0)

(𝑛 ≠ 0)

1. Mirror Chern number 2. Rotational Chern number

…etc (ignore weak TI here)

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### SnTe: mirror symmetry

T. H. Hsieh et al., Nature Commun. 3, 1192 (2012) S.-Y. Xu et al., Nature Commun. 3, 982 (2012) SnTe

+Γ(+) +

−−

−𝐿(−) +

−+

−L(−) +

−+

−X(+)

− ++

+X(+) +

−−

−L(+) +

−+

−L(+) +

−+

−Z(+)

− ++

EF Z2=0

Mirror Chern number

where 𝑛∓𝑖 = mirror eigenvalue

Even number of times band inversion + additional crystalline symmetry

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### TCI: mirror and glide mirror

T. H. Hsieh et al., Nature Commun. 3, 1192 (2012) S.-Y. Xu et al., Nature Commun. 3, 982 (2012)

SnTe => mirror KHgX(X=As,Sb,Bi) => glide mirror

Z. Wang et al., Nature 532, 189 (2016)

J. Ma et al., Science Advances 3, e1602415 (2017)

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New TCI: protected by the N-fold rotational symmetries Chen Fang and Liang Fu, arXiv:1709.01929 (2017).

C2 C4 C6

### Real material ???

Topological invariant number is not convenient to calculate from DFT.

Cn = n Dirac points on one surface

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### TCI: symmetry indicator

(symmetry indicator)

### How to implement on first-principle?

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Zhida Song, Tiantian Zhang, Zhong Fang, Chen Fang:

Nature Communications 9, 3530 (2018)

Systematic method (Fu-Kane-like formula) for searching TCI based on symmetry indicator

### TCI: symmetry indicator

when certain additional symmetry Y is present, topological invariants of TCIs protected by symmetry X can be inferred by the Y -symmetry eigenvalues of energy band.

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I4/mmm (#139) Ca2As: body-centered tetragonal lattice

Time-reversal Inversion

M(100) M(010) M(001) M(110) M(1-10)

C4(001) C2(100) C2(010) C2(001) C2(110) C2(1-10) Symmetry

operation

### TCI: candidate material Ca 2 As

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Energy (eV)

Counting the irred. rep. number of each band

The symmetry indicator of # 139 is (Z2,Z8) Step-1

### Ca 2 As: symmetry indicator

(Z2,Z8)=(0,4) for Ca2As

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Step-2 mirror glide mirror rotation

I4/mmm (#139)

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Step-3

### Ca 2 As: symmetry indicator

I4/mmm (#139)

mirror Chern number

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I4/mmm (#139)

C4 + Mirror

Energy (eV)

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### Topological surface states

I4/mmm (#139)

C4 + Mirror

C2 + Mirror

Energy (eV)

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Breaking mirror & C4

Breaking mirror

C2+Mirror C2

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Ca2As Ca2Bi

Sr2Sb

Phase transition

### Topological phase transition

TCI

TCI+Weak TI Normal insulator

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(1) Searching TCI by using symmetry indicator for the first time Ca2As (Z2,Z8)=(0,4)

(2) Topological surface states are protected by multi topological invariant numbers

(3) It is first time that the Topological surface states are protected by rotational symmetry

Breaking mirror

### Conclusion (TCI)

arXiv:1805.05215 (2018)

X. Zhou et al., Phys. Rev. B 98, 241104(R) (2018)

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### Nodal-line semimetal (節線半金屬)

Normal metal: 2D Fermi surface Topological metal: 1D Nodal-line

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NL

### Nodal-line semimetal

G. Bian et al., Nature Commun. 7, 10556 (2016) G. Bian et al., Phys. Rev. B 93, 12113(R) (2016)

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### Type-I vs Type-II

Type-II DP

Slope sign: opposite Slope sign:

the same

### Type-I

Slope sign: opposite Slope sign: opposite

Type-II Weyl: A. A. Soluyanov et al.,Nature volume 527, 495 (2016) Type-II Dirac: T.-R. Chang et al., Phys. Rev. Lett. 119, 026404 (2017)

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n-doping

### Mg 3 Bi 2

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Dr. Xiaoting Zhou (NCKU)

Chuang-Han Hsu (NUS)

Dr. Su-Yang Xu (MIT)

Prof. Liang Fu (MIT)

MOST(Taiwan) Young Scholar Fellowship:

MOST Grant for the Columbus Program

### Acknowledgements (TCI)

X. Zhou et al., Phys. Rev. B 98, 241104(R) (2018)

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### Acknowledgements (Nodal-line)

MOST(Taiwan) Young Scholar Fellowship:

MOST Grant for the Columbus Program Prof. R. J. Cava

(Princeton U.)

Prof. Weiwei Xie (Louisiana State U.)

Prof. Guang Bian

(University of Missouri)

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## Thank you !

To see a world in a grain of sand … －William Blake

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