Chern number and Z2 invariant
Hikaru Sawahata
Collabolators: Yo Pierre Mizuta, Naoya Yamaguchi, Fumiyuki Ishii
Graduate School of Natural Science and Technology, Kanazawa University

2016/11/25

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
1 / 27

Table of contents

1

Introduction
Chern insulator, Topological insulator
Topological invariant

2

Fukui-Hatsugai method(Chern number)
Z2 invariant

3

Test calculation and Application-Z2 invariant -Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
2 / 27

Introduction

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
3 / 27

Chern insulator, Topological insulator
Topological material: Chern insulator1 or Topological insulator2 has
special edge state(s).

12Thouless-Kohmoto et al.,Phys. Rev. Lett. 49, 405 (1982)
L. Fu and C. Kane, Phys. Rev. B 74, 195312 (2006)

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
4 / 27

Example of edge states
Bi(111) film’s(Topological insulator)3 band dispersion

3

S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
5 / 27

Topological invariant(1)Chern insulator is characterized by Chern number C defined as

Chern number
C=

"
occ.
!
1
n

2π

BZ

dkx dky Fn = 0, ±1, ±2, ±3, · · ·Fn = (∇ × An )z , An = i ⟨unk |∂|unk ⟩∂k

C = 0 corresponds to a trivial state andC = ±1, ±2, ±3, · · · corresponds to a Chern insulator state.
Topological insulator is characterized by Z2 invariant defined as

Z2 invariant
Z2 =

#$
occ.
!
1
n

2π

Half BZ

An · dk −"dkx dky Fn
Half BZ%= 0 or 1(mod 2)

Z2 = 0 corresponds to a trivial state andZ2 = 1 corresponds to a topological insulator state.

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
6 / 27

Topological invariant(2)Why is Topological invariant, Chern number or Z2 invariant important？
We can predict if special edge state(s) exist
by calculating Chern number or Z2 invariant in a bulk system→ NO need to calculate edge states.
Topological invariant, Chern number or Z2 invariant doesn’t change
without gap closing. This fact is analogous to mathematical ”Topology”

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
7 / 27

Fukui-Hatsugai method computing
Chern number and Z2 invariant

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
8 / 27

Chern number

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
9 / 27

Computing Chern number(1)-Overlap matrixThe definition of Chern number:"1Cn =dkx dky Fn = 0, ±1, ±2, · · ·2π BZ
we define overlap matrix U (we can calculate U by ”polB.c”)4
Uµ = det ⟨um (k)|un (k + ∆µ)⟩
and, we can obtain A and F as
Aµ (k) = Im log Uµ (k)Fkx ky (k) = Im log Ukx (k)Uky (k + ∆kx )Uk−1(k + ∆ky )Uk−1(k)xy
For obtaining Chern number, we need to calculate F on every ”tile” indiscretized Brillouin Zone and summate them 5 .
45Electric Polarization by Berry Phase:Ver. 1.1, Technical Notes on OpenMX
T. Fukui, Y. Hatsugai and H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005).

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
10 / 27

Computing Chern number(2)-Calculate FDiscretized Brillouin Zone in the directionof kx and ky , and calculate on every ”tile”,
U12 = det ⟨u1 |u2 ⟩U23 = det ⟨u2 |u3 ⟩U34 = det ⟨u3 |u4 ⟩U41 = det ⟨u4 |u1 ⟩Fk = Im log U12 U23 U34 U41

and, we can obtain Chern number as
C=

occ.
!
n

BZ

Cn =

1 !
Fk
2π
k

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
11 / 27

Z2 invariant

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
12 / 27

The method of computing Z2 invariant
There are three methods for computing Z2 invariant
MethodParity6
WFC78
Fukui-Hatsugai9

Advantage
Easy
intuitive
DirectDisadvantage
Only inversion symmetric system
indirect
not intuitive

We implemented these three methods.
In this presentation, we introduce a Fukui-Hatsugai method.
Fukui-Hatsugai is direct method and suitable for automatic searching for
topological insulators. (application to material informatics)
6

L. Fu and C. Kane, Phys. Rev. B 76, 045302 (2007).
A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B 83, 235401 (2011).
8
R. Yu, X. Qi, A. Bernevig, Z. Fang, and X. Dai, Phys. Rev. B 84, 075119 (2011).
9
T.Fukui and Y.Hatsugai, J. Phys. Soc. Jpn. 76, 053702 (2007).
7

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
13 / 27

Computing Z2 invariant -Fukui-Hatsugai methodComputing Z2 invariant by Fukui-Hatsugai method is almost same ascomputing Chern number.
%"occ. #$1 !
Z2=An · dk −Fn dkx dky (mod 2)
2π nHalf BZ
Half BZ
U12 = det ⟨u1 |u2 ⟩U23 = det ⟨u2 |u3 ⟩U34 = det ⟨u3 |u4 ⟩U41 = det ⟨u4 |u1 ⟩Fk = Im log U12 U23 U34 U41
Aab = Im log Uab
1n(k) =[(A12 + A23 + A34 + A41 ) − Fk ]2π
Half BZ
1 !
Z2 =n(k) (mod 2)2πkHikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
14 / 27

Test calculation and Application-Z2 invariant -Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
15 / 27

How to use

we made two codes, ”calB.c”(Chern number) and ”Z2FH.c”(Z2 invariant)by modifying ”polB.c” (calculate polarization by Berry phase)So, you have to prepare (system name).scfout fileby keyword specified in input file
HS.fileoutonand you can obtain Chern number or Z2 invariant by(PATH)/calB(PATH)/Z2FH(system name).scfout(system name).scfout

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
16 / 27

Test calculation-Z2 invariant (1)Z2 invariant calculated by Fukui-Hatsugai method which is implemented in
OpenMX, completely agrees for all these materials.
Material
Bi2 Te3
Bi2 Se3
Sb2 Te3Sb2 Se3
Bi(111) film
TlBiTe2
TlBiSe2
TlBiS2
TlSbTe2
TlSbSe2
TlSbS2

MethodParityParityParityParityParityParityParityParityParityParityParityZ2
1;(000)1;(000)1;(000)0;(000)
1
1;(000)1;(000)0;(000)1;(000)1;(000)0;(000)

Fukui-Hatsugai
1;(000)1;(000)1;(000)0;(000)
1
1;(000)1;(000)0;(000)1;(000)1;(000)0;(000)Previous study
[1]
[1]
[1]
[1]
[2]
[3]
[3]
[3]
[3]
[3]
[3]

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
17 / 27

Test calculation-Z2 invariant (2)Z2 invariant calculated by Fukui-Hatsugai method which is implemented in
OpenMX, completely agrees for all these materials.
Material
YBiO3
YSbO3
AlBi,zincblendeBBi,zincblendeGaBi,zincblendeInBi,zincblende
TlSbX2 film
CsPbI3
CsPbI3 , FE

MethodParityParity
WFC
WFC
WFC
WFC
WFC
WFC
WFC

Z2
1;(111)
1;(111)
1;(000)1;(000)1;(000)1;(000)11;(111)
1;(111)

Fukui-Hatsugai
1;(111)
1;(111)
1;(000)1;(000)1;(000)1;(000)11;(111)
1;(111)

Previous study
[4]
[4]
[5]
[5]
[5]
[5]
[6]
[7]
[7]

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
18 / 27

[1] H. Zhang, C. Liu, X. Qi, X. Dai, Z. Fang, and S. Zhang,Nat. Phys. 5,438 (2009).
[2] S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).
[3] B. Singh et al., Phys. Rev. B. 86, 115208 (2012).
[4] H. Jin, S. Rhim, J. Im, and A. Freeman, Sci. Reports 3, 1651 (2013).
[5] H. Huang et al., Phys. Rev. B 90, 195105 (2014).
[6] R. W. Zhang et al. Sci. Reports 6, 21351 (2016).
[7] S. Liu, Y. Kim, L. Tan, and A. Rappe, Nano. Lett. 16, 1663 (2016).

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
19 / 27

Application(1)-Z2 invariantWe predict topological phase transition on Bi(111) film by electric field,
topological insulator to normal insulator.

Metal

TI Phase

NI Phase

Z2 = 1Z2 = 0Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
20 / 27

Application(2)-Z2 invariantWe confirmed Z2 invariant by WFC and Fukui-Hatsugai.
E = 2.5 V/Å

Hatsugai

0.5

0.5

0.43750.4375

0.375

0.375

0.31250.31250.25

0.25

0.18750.1875

0.125

0.125

0.06250.0625ky (2//a)Fukui-ky (2//a)

E = 0 V/Å

0-0.06250-0.0625-0.125-0.125-0.1875-0.1875

-0.25

-0.25

-0.3125-0.3125-0.375-0.375-0.4375-0.4375-0.5-0.5-0.4375 -0.375 -0.3125 -0.25 -0.1875 -0.125 -0.0625

00.0625 0.125 0.1875

kx (2//a)0.25

0.3125 0.375 0.4375

0.5

-0.5
-0.5

-0.4375 -0.375 -0.3125 -0.25 -0.1875 -0.125 -0.0625

00.0625 0.125 0.1875

0.25

0.3125 0.375 0.4375

0.5

kx (2//a)WFC

Z2 = 1Z2 = 0Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
21 / 27

Summary

Chern insulator or topological insulator has special edge states.
Chern insulator is characterized ”Chern number” and topologicalinsulator is characterized ”Z2 invariant”.
These values are obtained by ”Fukui-Hatsugai method”(calculation of
Berry curvature F and Berry connection A by overlap matrix).
This method is direct, so this is suitable for automatic searching formaterial, topological insulators or Chern insulators,

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
22 / 27

Appendix

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
23 / 27

Quantum Hall eﬀect, Quantum spin Hall eﬀect

Hall conductivity σxy
described by Chern number as
σxy = Ce2hMaterial
Symmetry
Edge state
Topological invariant

Chern insulator
Charge current
Chern number

Topological insulator
Time reversal symmetry
Spin current
Z2 invariant

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
24 / 27

Calculate Z2 invariant(1)-Calculate F and AThe definition of Z2 invariant:%"occ. #$1 !
Z2=An · dk −Fn dkx dky (mod 2)
2π nHalf BZ
Half BZ
Computing Z2 invariant is same as Chern number.
Calculate A and F by overlap matrix every ”tile” and calculate n(k) which
is defined,n(k)1[Ak (k + ∆kx ) − Aky (k)2π y−(Akx (k + ∆ky ) − Akx (k)) − Fkx ky (k)]=−n(k) is called ”Lattice Chern number”, we can obtain Z2 invariant as,
1
Z2 =
2π

Half
!BZ

n(k) (mod 2)kHikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
25 / 27

Calculate Z2 invariant(2)-Lattice Chern numberDiscretized Brillouin Zone in the directionof kx and ky , and calculate on every ”tile”,
U12 = det ⟨u1 |u2 ⟩U23 = det ⟨u2 |u3 ⟩U34 = det ⟨u3 |u4 ⟩U41 = det ⟨u4 |u1 ⟩Fk = Im log U12 U23 U34 U41
Aab = Im log Uaband, we can obtain Lattice Chern number asn(k) =

Red circle is +1, blue triangle
is -1, blank is 0

1[(A12 + A23 + A34 + A41 ) − Fk ]2π

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
26 / 27

Calculate Z2 invariant(3)-Gauge fixWhen we calculate Z2 invariant, we summate only Half BZ, so summation
is gauge dependent!! →we have to fix gauge
Red point · · ·Translation symmetricpointsBlue point· · ·Time reversal symmetricpointsYellow point· · ·Kramars degenerate
pointsI’ll explain the details of fixing gauge if extra time is available

Hikaru SawahataCollabolators: Yo Pierre Mizuta, Naoya
Chern
Yamaguchi,
number andFumiyuki
Z2 invariant
Ishii (Graduate School of Natural
2016/11/25
Science and Technology
27 / 27