トポロジカル絶縁体・超伝導体

古崎

昭

（理化学研究所）

topological insulators (3d and 2d)July 9, 2009

Outline• イントロダクション

バンド絶縁体,  トポロジカル絶縁体・超伝導体

• トポロジカル絶縁体・超伝導体の例:

整数量子ホール系

p+ip 超伝導体

Z2

トポロジカル絶縁体: 2d & 3d

• トポロジカル絶縁体・超伝導体の分類

絶縁体 Insulator

• 電気を流さない物質

絶縁体

Band theory of electrons in solids

• Schroedinger equation

ion (+ charge)

electron Electrons moving in the lattice of ions

( ) ( ) ( )rErrVm

ψψ =⎥⎦

⎤⎢⎣

⎡+∇− 2

2

2h

( ) ( )rVarV =+

Periodic electrostatic potential from ions and other electrons (mean‐field)

Bloch’s theorem:( ) ( ) ( ) ( )ruaru

ak

aruer knknkn

ikr,,, , , =+<<−=

ππψ

( )kEn Energy band dispersion :n band index

例：一次元格子模型

ε2A A A A AB B B B B

t−

n 1+n1−n

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

B

Aikn

uu

enψ ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−−

B

A

B

A

uu

Euu

ktkt

εε

)2/cos(2)2/cos(2

222 )2/(cos4 ε+±= ktE

k

E

ππ− 0

ε2

Metal and insulator in the band theory

k

E

aπ

aπ− 0

Interactions between electronsare ignored. (free fermion)

Each state (n,k) can accommodateup to two electrons (up, down spins).

Pauli principle

Band Insulator

k

E

aπ

aπ− 0

Band gap

All the states in the lower band are completely filled. (2 electrons per unit cell)

Electric current does not flow under (weak) electric field.

ε2

ε2

Topological (band) insulators

Condensed-matter realization of domain wall fermions

Examples: integer quantum Hall effect,

quantum spin Hall effect, Z2

topological insulator, ….

• バンド絶縁体

• トポロジカル数をもつ

• 端にgapless励起(Dirac fermion)をもつstable

Topological (band) insulators

Condensed-matter realization of domain wall fermions

Examples: integer quantum Hall effect,

quantum spin Hall effect, Z2

topological insulator, ….

• バンド絶縁体

• トポロジカル数をもつ

• 端にgapless励起(Dirac fermion)をもつ

Topological superconductorsBCS超伝導体

(Dirac or Majorana) をもつ

p+ip superconductor, 3He

stable

Example 1: Integer QHE

Prominent example: quantum Hall effect

• Classical

Hall effect

Br

e− −−−−−−

++++++

Er

vr

W

:n electron density

Electric current nevWI −=

BcvE =Electric field

Hall voltage IneBEWVH −

==

Hall resistanceneBRH −

=

Hall conductanceH

xy R1

=σBveFrrr

×−=Lorentz force

Integer quantum Hall effect

(von Klitzing 1980)

Hxy ρρ =

xxρ Quantization of Hall conductance

heixy

2

=σ

Ω= 807.258122eh

exact, robust against disorder etc.

Integer quantum Hall effect• Electrons are confined in a two-dimensional plane.

(ex. AlGaAs/GaAs interface)

• Strong magnetic field is applied(perpendicular to the plane)

Landau levels:

( ) ,...2,1,0 , ,21 ==+= n

mceBnE ccn ωωh

AlGaAs GaAsBr

cyclotron motion

k

E

ππ−

TKNN number  (Thouless‐Kohmoto‐Nightingale‐den Nijs)TKNN (1982); Kohmoto (1985)

Che

xy

2

−=σ

Chern number (topological invariant)

∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

−∂∂

∂∂

=yxxy k

uku

ku

kurdkd

iC

**22

21π

( )yxk kkAkdi

, 21 2

rr×∇= ∫π

filled band

( ) kkkyx uukkA rrrr∇=,

integer valued

)(rue krki r

rrr⋅=ψ

Topological insulators: band insulators characterized by a topological number

Best known example:IQHE in 2DEG

Quantized Hall conductance

he

xy

2

×Ζ∈σ

Edge states

GS0=xyσ 1+=xyσ 2+=xyσ2−=xyσ 1−=xyσ

Quantum phase transitions

Space of gapped groundstates is partitioned intotopological sectors.

TKNN

Ζ∈#stable against disorder

Edge states• There is a gapless chiral edge mode along the sample

boundary.Br

xk

E

ππ−Number of edge modes C

hexy =

−=

/2

σ

Robust against disorder (chiral fermions cannot be backscattered)

Bulk: (2+1)d Chern-Simons theoryEdge: (1+1)d CFT

( )xm

x

Effective field theory

( ) ( ) zyyxx xmivH σσσ +∂+∂−=

( ) zyyxx mivH σσσ +∂+∂−=

parity anomaly ( )mxy sgn21

=σ

Domain wall fermion

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎥⎦

⎤⎢⎣⎡ −= ∫ i

dxxmv

ikyyxx

y

1''1exp,

0σψ

vkE −= 0>m 0<m

Example 2: chiral p-wave superconductor

BCS理論

（平均場理論）

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ]∑∫

∑∫

↓=↑

++

↓=↑

+

Δ+Δ+

⎥⎦

⎤⎢⎣

⎡−−∇−=

,,

*,,

,

22

'','','21

2

βαβαβαβαβα

σσσ

ψψψψ

ψμψ

rrrrrrrrrrdd

rAiem

rrdH

dd

drrh

( ) ( ) ( ) ( )''',, rrrrVrr βαβα ψψ−=Δ 超伝導秩序変数

S‐wave (singlet) ( ) ( ) ( ) ( )rrrrr Ψ↑↓−↓↑−=Δ 2

1'',, δβα

0ξ

P‐wave (triplet)

+

( ) ( ) ( )rrrr

rr Ψ↑↑−∂∂

=Δ ''

',, δβα

xkk kk 0)( Δ∝∝Δ −ψψr

高温超伝導体22)( yxkk kkk −∝∝Δ −ψψ

r22 yx

d−

Integrating outGinzburg‐Landau

σψ

Spinless

px

+ipy

superconductor in 2 dim.• Order parameter

• Chiral (Majorana) edge state

)()( 0 yxkk ikkk +Δ∝∝Δ −ψψr

k

E

Δ

Ψ⋅Ψ=Ψ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−Δ+Δ−

Ψ= ++ 21

2)()()(2)(

21

22

22

σrr

kFFyx

FyxF hmkkkikk

kikkmkkH

Hamiltonian density ⎟⎟⎠

⎞⎜⎜⎝

⎛=Ψ +

−k

k

ψψ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+Δ−

Δ=

mkkk

kk

kkh Fyx

F

y

F

xk 2

222r

khEr

±=Gapped fermion spectrum22 : SS

hh

k

k →r

r

1=zL

( ) ( )[ ]ψψψψψμψ yxyxF

iik

im

H ∂−∂+∂+∂Δ

−⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−= +++

222

2h

( )( )

( )( )⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+ rv

rue

trtr iEt h/

,,

ψψ

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−∂−∂Δ−

∂+∂Δ−

vu

Evu

hiiiih

yx

yx

0

0

Hamiltonian density

Bogoliubov-de Gennes equation

⎟⎟⎠

⎞⎜⎜⎝

⎛→⎟⎟

⎠

⎞⎜⎜⎝

⎛−→ *

*

,uv

vu

EE Particle-hole symmetry (charge conjugation)

zeromode: Majorana fermion

[ ]Ht

i ,ψψ=

∂∂

Majorana edge state

( ) ( )

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+∂+∂∂−∂Δ

−

∂+∂Δ

−+∂+∂−

vu

Evu

km

ik

i

ik

ikm

FyxyxF

yxF

Fyx

222

222

21

21

Δ<E

px

+ipy

superconductor: vacuum:0>y 0<y 0=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

vu

y

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ+=⎟⎟

⎠

⎞⎜⎜⎝

⎛11

cosexp 22 ykkykmikx

vu

FFk

k

FkkE Δ=

k

E

Δ2

( ) ( ) ( )( )∫ +Δ−−Δ− +=F

FF

k

kktxik

kktxik eedktx

0

//

4, γγ

πψ

+=ψψ

( ) ( )∫∫ ∂Δ

−=Δ

= + yydyk

ikkdkH y

F

k

kkF

F

ψψγγ0

edge

• Majorana bound state in a quantum vortex

vortex ehc

=φ

Fn En / , 2000 Δ≈= ωωε

Bogoliubov-de Gennes equation

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−ΔΔ

− vu

vu

heeh

i

i

εϕ

ϕ

*0

0 ( ) FEAepm

h −+=2

0 21 rr

zero mode

phase⎟⎟⎠

⎞⎜⎜⎝

⎛⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛− 2/

2/

ϕ

ϕ

i

i

veue

vu

⎟⎟⎠

⎞⎜⎜⎝

⎛⇔⎟⎟

⎠

⎞⎜⎜⎝

⎛ΨΨ+ v

u

00 =ε )( γ=Ψ=Ψ + Majorana (real) fermion!

ϕ

energy spectrum of vortex bound states

2N vortices GS degeneracy = 2N

phase winding:π2 γγ −→

interchanging vortices braid groups, non-Abelian statistics

i i+11+→ ii γγ

ii γγ −→+1

D.A. Ivanov, PRL (2001)

Fractional quantum Hall effect at 

• 2nd

Landau level• Even denominator (cf. Laughlin states: odd denominator)• Moore-Read (Pfaffian) state

25

=ν

jjj iyxz +=

( ) ∑−Π⎟⎟⎠

⎞⎜⎜⎝

⎛

−= −

<

4/2MR

21Pf izjiji

ji

ezzzz

ψ ( ) ijij AA det Pf =

Pf(    ) is equal to the BCS wave function of px

+ipy

pairing state.

Excitations above the Moore-Read state obey non-Abelian statistics.

Effective field theory: level-2 SU(2) Chern-Simons theory

G. Moore & N. Read  (1991); C. Nayak & F. Wilczek (1996)

Example 3: Z2

topological insulator

Quantum spin Hall effect

Quantum spin Hall effect (Z2

top. Insulator)

• Time-reversal invariant band insulator• Strong spin-orbit

interaction

• Gapless helical edge mode (Kramers pair)

Kane & Mele (2005, 2006); Bernevig & Zhang (2006)

Br

Br

−

up-spin electrons

down-spin electrons

σλ rr⋅L

No spin rotation symmetry

kykx

E

K

K’

K’

K’

K

K

Kane-Mele model (PRL, 2005)

( )∑ ∑ ∑∑ ++++ +×++=ij ij i

iiivij

jzijiRjz

iijSOji cccdscicscicctH ξλλνλrr

t

SOiλ

SOiλ−

i j

ijdr

A B( ) ( )( ) ↓=↑=Ψ ⋅ , ,, skke sBsA

rkirrrr

φφ

( ) ( ) zv

yxxyR

zzSOy

yx

xK sssivHK σλσσλσλσσ +−+−∂+∂=

2333 :

(B) 1 (A), 1 −+=iξ

( ) ( ) zv

yxxyR

zzSOy

yx

xK sssivHK σλσσλσλσσ ++++∂+∂−=

2333 :' '

'*

Ky

Ky HisHis =− time reversal symmetry

• Quantum spin Hall insulator is characterized by Z2

topological index ν1=ν an odd

number of helical edge modes; Z2

topological insulator

an even (0)

number of helical edge modes0=ν1 0

Kane-Mele modelgraphene + SOI[PRL 95, 146802 (2005)]

Quantum spin Hall effectπ

σ2es

xy =

(if  Sz

is conserved)

0=Rλ

Edge states stable against disorder (and interactions)

Z2

: stability of gapless edge states

(1) A single Kramers doublet

zzyyx

xxz sVsVsVVivsH ++++∂= 0 HisHis yy =− *

(2) Two Kramers doublets

( ) ( )zzyyx

xy

xz sVsVsVVsIivH ++⊗++∂⊗= τ0

remain to be gapless

opens a gap

Odd number of Kramers doublet (1)

Even number of Kramers doublet (2)

ExperimentHgTe/(Hg,Cd)Te quantum wells

Konig et al.  [Science 318, 766 (2007)]

CdTeHgCdTeCdTe

Example 4: 3-dimensional Z2

topological insulator

3‐dimensional Z2

topological insulatorMoore & Balents; Roy; Fu, Kane & Mele (2006, 2007)

bulkinsulator

surface Dirac fermion(strong) topological insulatorbulk: band insulatorsurface: an odd

number of surface Dirac modes

characterized by Z2

topological numbers

Ex: tight‐binding model with SO int. on the diamond lattice[Fu, Kane, & Mele; PRL 98, 106803 (2007)]

trivial insulator

Z2

topologicalinsulator

trivial band insulator: 0 or an even

number of 

surface Dirac modes

Surface Dirac fermions

• A “half”

of graphene

• An odd number of Dirac fermions in 2 dimensionscf. Nielsen-Ninomiya’s no-go theorem

kykx

E

K

K’

K’

K’

K

K

topologicalinsulator

Experiments• Angle-resolved photoemission spectroscopy (ARPES)

Bi1‐x

Sbx

p, Ephoton

Hsieh et al., Nature 452, 970 (2008)

An odd (5) number of surface Dirac modes were observed.

Experiments II

Bi2

Se3

Band calculations (theory)

Xia et al.,Nature Physics 5, 398 (2009)

“hydrogen atom”

of top. ins.

a single Dirac cone

ARPES experiment

トポロジカル絶縁体・超伝導体の分類

Schnyder, Ryu, AF, and Ludwig,  PRB 78, 195125 (2008)

arXiv:0905.2029 (Landau100)

Classification oftopological insulators/superconductors

Schnyder, Ryu, AF, and Ludwig, PRB (2008)

Kitaev,  arXiv:0901.2686

Zoo of topological insulators/superconductors

Topological insulators are stable against (weak) perturbations.

Classification of topological insulators/SCs

Random deformation of Hamiltonian

Natural framework: random matrix theory (Wigner, Dyson, Altland & Zirnbauer)

Assume only basic discrete symmetries:

(1) time-reversal symmetry

HTTH =−1*0     no TRS

TRS =      +1   TRS with‐1    TRS with

TT +=Τ

TT −=Τ(integer spin)

(half‐odd integer spin)

(2) particle-hole symmetry

HCCH −=−Τ 10     no PHS

PHS =      +1   PHS with‐1    PHS with

CC +=Τ

CC −=Τ(odd parity: p‐wave)

(even parity: s‐wave)

( ) HTCTCH −=−110133 =+×

(3) TRS PHS chiral symmetry [sublattice symmetry (SLS)]× =

yisT =

(2) particle-hole symmetry Bogoliubov-de Gennes

( ) ( ) zkyyxxkk

kkkk kkh

cc

hccH σεσσ ++Δ=⎟⎟⎠

⎞⎜⎜⎝

⎛= +

−−

+ 21

px

+ipy

Txkxkx CChh ==−=− σσσ *

dx2-y2

+idxy

( ) ( )[ ] zkyyyxyxkk

kkkk kkkkh

cc

hccH σεσσ ++−Δ=⎟⎟⎠

⎞⎜⎜⎝

⎛= +

↓−

↑↓−

+↑

22 21

Tykyky CiChh ==−=− σσσ *

10 random matrix ensembles

IQHE

Z2

TPI

px

+ipy

Examples of topological insulators in 2 spatial dimensionsInteger quantum Hall EffectZ2

topological insulator (quantum spin Hall effect) also in 3DMoore-Read Pfaffian state (spinless p+ip superconductor)

dx2‐y2

+idxy

Table of topological insulators in 1, 2, 3  dim. Schnyder, Ryu, Furusaki & Ludwig, PRB (2008)

Examples:(a)Integer Quantum Hall Insulator,  (b) Quantum Spin Hall Insulator,(c)  3d Z2

Topological Insulator,  (d)  Spinless chiral p‐wave (p+ip) superconductor (Moore‐Read),(e)Chiral d‐wave                            superconductor,  (f)            

superconductor,

(g)  3He B phase.

Classification of 3d topological insulators/SCs

strategy (bulk boundary)• Bulk topological invariants

integer topological numbers：

3 random matrix ensembles

(AIII, CI, DIII)

• Classification of 2d Dirac fermions

Bernard & LeClair  (‘02)

13

classes

(13=10+3)        AIII, CI, DIII             AII, CII

• Anderson delocalization in 2dnonlinear sigma models

Z2

topological term (2)

or

WZW term (3)

BZ: Brillouin zone

Topological distinction of ground states

m filledbands

n

empty

bands

xk

ykmap from BZ to Grassmannian

( ) ( ) ( )[ ] Ζ=×+ nUmUnmU2π IQHE (2 dim.)

homotopy class

deformed “Hamiltonian”

In classes AIII, BDI, CII, CI, DIII, Hamiltonian can be made off‐diagonal.

Projection operator is also off‐diagonal.

( )[ ] Ζ=nU3π topological insulators labeled by an integer

[ ] ( )( )( )[ ]∫ ∂∂∂= −−− qqqqqqkdq νμλλμνε

πν 111

2

3

tr24

Discrete symmetries limit possible values of [ ]qν

Z2

insulators in CII (chiral symplectic)

The integer number                    # of surface Dirac (Majorana) fermions( )qν

bulkinsulator

surfaceDiracfermion

(3+1)D 4-component Dirac Hamiltonian

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

−⋅⋅

=+⊗=mk

kmmkkH x σστστ μμμ rrrr

AII: ( ) ( )kHikHi yy −=− σσ * TRS

DIII: ( ) ( )kHkH yyyy −−=⊗⊗ στστ * PHS

AIII: ( ) ( )kHkH yy −=ττ chS

[ ] ( )mq sgn21

=ν

( )zm

z

(3+1)D 8-component Dirac Hamiltonian⎟⎟⎠

⎞⎜⎜⎝

⎛= + 0

0D

DH

( ) ( ) ( )imkikikD yyy −⊗−=−= μμμμ σστγαβσ 5CI:

( ) ( )kDkDT −=

CII: ( ) ( )kDmkkD +=+= βαμμ ( ) ( )kDikDi yy −=− σσ *

[ ] ( ) 2sgn21

×= mqν

[ ] 0=qν

( )zm

z

Classification of 3d topological insulators

strategy (bulk boundary)

• Bulk topological invariants

integer topological numbers：

3 random matrix ensembles

(AIII, CI, DIII)

• Classification of 2d Dirac fermions

Bernard & LeClair  (‘02)

13

classes

(13=10+3)        AIII, CI, DIII             AII, CII

• Anderson delocalization in 2dnonlinear sigma models

Z2

topological term (2)

or

WZW term (3)

Nonlinear sigma approach to Anderson localization• (fermionic) replica• Matrix field Q describing diffusion• Localization massive• Extended or critical massless topological Z2

termor WZW term

Wegner, Efetov, Larkin, Hikami, ….

( ) 22 ZM =π

( ) ZM =3π

Table of topological insulators in 1, 2, 3  dim. Schnyder, Ryu, Furusaki & Ludwig, PRB (2008)

arXiv:0905.2029

Examples:(a)Integer Quantum Hall Insulator,  (b) Quantum Spin Hall Insulator,(c)  3d Z2

Topological Insulator,  (d)  Spinless chiral p‐wave (p+ip) superconductor (Moore‐Read),(e)Chiral d‐wave                            superconductor,  (f)            

superconductor,

(g)  3He B phase.

Reordered TableKitaev, arXiv:0901.2686

Periodic table of topological insulators

Classification in any dimension

Classification oftopological insulators/superconductors

Schnyder, Ryu, AF, and Ludwig, PRB (2008)

Kitaev,  arXiv:0901.2686

Summary• Many topological insulators of non-interacting fermions

have been found.interacting fermions??

• Gapless boundary modes (Dirac or Majorana)stable against any (weak) perturbation disorder

• Majorana

fermionsto be found experimentally in solid-state devices

Z2

T.I. + s-wave SC Majorana

bound state (Fu & Kane)

Z2

T.I. + test charge Dyon

(Qi, Li, Zang, & S.-C. Zhang)

∫ ⋅= BExdtdhceS

rr3

2

2πθ

θ