Introduction to topological
insulators and superconductors
古崎 昭 （理化学研究所）
September 16, 2009
topological insulators (3d and 2d)
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
ion (+ charge)
electron
Electrons moving in the lattice of ions
• Schroedinger equation
 2 2

  V r  r   E r 

 2m

Bloch’s theorem:
 r   e un ,k r , 
ikr

a
k
En k  Energy band dispersion

a
Periodic electrostatic potential from
ions and other electrons (mean-field)
V r  a   V r 
, un ,k r  a   un ,k r 
n : band index
例：一次元格子
t
2
A
B
A
n 1
B
n
uA 
 n   e  
 uB 
ikn
A
B
A
B
A
B
n 1

 2t cos(k / 2)  u A 
uA 


   E  

  2t cos(k / 2)
 u B 
 uB 
E
E   4t 2 cos 2 (k / 2)   2
2

0

k
Metal and insulator in the band theory
E
Interactions between electrons
are ignored. (free fermion)
Each state (n,k) can accommodate
up to two electrons (up, down spins).

a
0

k
a
Pauli principle
Band Insulator
E
Band gap 2

a
0

k
a
All the states in the lower band are completely filled.
(2 electrons per unit cell)
2
Electric current does not flow under (weak) electric field.
Topological (band) insulators
• バンド絶縁体
free fermions
• トポロジカル数をもつ
• 端にgapless励起(Dirac fermion)をもつ
stable
Condensed-matter realization of domain wall fermions
Examples: integer quantum Hall effect,
quantum spin Hall effect, Z2 topological insulator, ….
Topological (band)
insulators
superconductors
• BCS超伝導体
バンド絶縁体
超伝導gap
• トポロジカル数をもつ
• 端にgapless励起(Dirac
(Dirac fermion)をもつ
or Majorana) をもつ
stable
Condensed-matter realization of domain wall fermions
Examples: p+ip superconductor, 3He
Example 1: Integer QHE
Prominent example: quantum Hall effect
• Classical Hall effect

B








v
E
e






n : electron density
Electric current
I  nevW
Electric field
v
E B
c
Hall voltage
W
Lorentz force

 
F  ev  B
VH  EW 
B
I
 ne
B
 ne
1

RH
Hall resistance
RH 
Hall conductance
 xy
Integer quantum Hall effect
(von Klitzing 1980)
 xy   H
h
 25812 .807 
2
e
 xx
Quantization of Hall conductance
 xy
e2
i
h
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)

B
AlGaAs
GaAs
E
cyclotron motion
Landau levels:
En  c n  ,
1
2
eB
c 
, n  0,1,2,...
mc


k
TKNN number
 xy  
(Thouless-Kohmoto-Nightingale-den Nijs)
TKNN (1982); Kohmoto (1985)
2
e
C
h
Chern number
(topological invariant)

ik  r

  e u k (r )
*
*

1

u

u

u
u 
2
2 
C
d k d r

integer valued



2i filled band
 k y k x k x k y 




1
2

d k  k  Ak x , k y 
Ak x , k y   uk  k uk

2i
Edge states
• There is a gapless edge mode along the sample boundary.

B
Number of edge modes 
  xy
2
e /h
C
Robust against disorder (chiral fermions cannot be backscattered)
Bulk: (2+1)d Chern-Simons theory
Edge: (1+1)d CFT
Effective field theory
H  iv x x   y  y   m z
parity anomaly
 xy  sgn m 
1
2
mx 
Domain wall fermion
H  iv x x   y  y  mx z
x
x
1

 1 
 x, y   expiky   y  mx'dx' 
0
v

  i 
E  vk
m0
m0
Example 2:
chiral p-wave superconductor
Integrating out  
Ginzburg-Landau
BCS理論 （平均場理論）


2
 2 

H   d r    r 
  ieA     r 
  ,
 2m

1
  d d rd d r '   ,  r , r '  r   r '  * ,  r , r '  r   r '
2
 ,   ,

d


 ,  r , r '  V r  r '   r   r '
S-wave (singlet)
0
P-wave (triplet)
+
超伝導秩序変数


1
 , r , r '   r  r '
     r 
2
高温超伝導体
d x2  y 2

(k )   k k  k x2  k y2

 ,  r , r ' 
 r  r '    r 
r '

(k )   k k   0 k x  ik y 
Spinless px+ipy superconductor in 2 dim.

• Order parameter (k )   k k  0 (k x  ik y )
• Chiral (Majorana) edge state
Lz  1
E

k
Hamiltonian density
2
2

(
k

k
1
F ) 2m

H k   
2
 (k x  ik y ) k F
  k
hk   x
 k
 F

k y
kF
k 
    
 k 
 
(k x  ik y ) k F 
1

   hk   
2
2
 (k  k F ) 2m 
2

(wrapping)
hk Gapped
fermion
spectrum
2
2 winding
k x2  k y2  k F2 

 : S  S number=1


2m
hk E   hk

Hamiltonian density
2





2
H    
     i
   x  i y     x  i y 
2k F
 2m


Bogoliubov-de Gennes equation
h0
 i x  i y  u 

u 

   E 
  i  i 
 v

h
x
y
0
v

 
 u   v* 
E   E ,     * 
 v  u 


i
  , H 
t
  r , t   iEt /   ur 
 
  e


 r , t 
 vr 
Particle-hole symmetry (charge conjugation)
zeromode: Majorana fermion
Majorana edge state E  
px+ipy superconductor:

 1 2
 x   2y  k F2

 2m


  i k  x  i y 
F


y0

 i  x  i y 
kF
1 2
 x   2y  k F2
2m



 u   E  u 
v
 v 
 




k
E
kF
 u 
   0
y  v 


 uk 
1
m 
2
2
   exp ikx 
y  cos k F  k y  
kF 
1
 vk 

kF
dk ik  x t / k F 
 x, t   
e
 k  e ik  x t / k F  k
4
0
  

E
2
k
kF
H edge   dk
0
k 

 k  k  i  dy  y  y  y 
kF
kF

• Majorana bound state in a quantum vortex
vortex  
hc
e
Bogoliubov-de Gennes equation
 h0
 i
 e

ei  u   u 
     
*  
 h0  v   v 


2
1 
h0 
p  eA  EF
2m
   u 
     
  v
energy spectrum of vortex bound states
 n  n0 , 0  20 / EF
zero mode  0  0
 0   0
Majorana (real) fermion!
2N vortices
GS degeneracy = 2N
interchanging vortices
i
i+1
braid groups, non-Abelian statistics
 i   i 1
 i 1   i
D.A. Ivanov, PRL (2001)
Fractional quantum Hall effect at
5

2
• 2nd Landau level
• Even denominator (cf. Laughlin states: odd denominator)
• Moore-Read (Pfaffian) state
z j  x j  iy j
 MR
 1
 P f
z z
j
 i
2

  zi  z j 2 e  zi / 4
 i j

Pf Aij   det Aij
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)
Kane & Mele (2005, 2006); Bernevig & Zhang (2006)
• Time-reversal invariant band insulator
 
  
• Strong spin-orbit interaction L  S  p  E  S
• Gapless helical edge mode (Kramers pair)


B

up-spin electrons

B
down-spin electrons
If Sz is conserved,
 xycharge   xy,   xy,  0
 xyspin   xy,   xy,  2 xy,
If Sz is NOT conserved,
Chern # (Z)
Z2
Quantized spin Hall conductivity
(trivial)
Band insulator
Quantum Hall
state
Quantum Spin
Hall state
Kane-Mele model
(PRL, 2005)

i  1 (A), 1 (B)

 
H  t  c c j  iSO  c s c j  iR  c s  d ij z c j  v   i ci ci

i
 z
ij i
ij

i
ij
ij
i
t
K
K’
K

d ij
 iSO
A
j
i
B

e


ik  r
ky
kx
K’
K
iSO
E
K’
  
A, s


k
B ,s
 

k

s  , 
3
K : H K  iv   x    y  3 3SO s  R  y s x   x s y  v z
2
3
x
y
z z
K ': H K '  iv    x    y  3 3SO s  R  y s x   x s y  v z
2
x

 is y H K* is y  H K '
y
z
z

time reversal symmetry


• Quantum spin Hall insulator is characterized by
Z2 topological index 
 1
 0
an odd number of helical edge modes; Z2 topological insulator
an even (0) number of helical edge modes
1
0
R  0
Kane-Mele model
graphene + SOI
[PRL 95, 146802 (2005)]
Quantum spin Hall effect
(if Sz is conserved)
Edge states stable against disorder (and interactions)
 xys 
e
2
Z2 topological number
Z2: stability of gapless edge states
(1) A single Kramers doublet
H  ivsz  x  V0  Vx s x  Vy s y  Vz sz
 is y H *is y  H
Kramers’ theorem
(2) Two Kramers doublets



H  iv I  s z  x  V0  y  Vx s x  Vy sy  Vz sz
opens a gap
Odd number of Kramers doublet
(1)
Even number of Kramers doublet
(2)

Experiment
HgTe/(Hg,Cd)Te quantum wells
CdTe
HgCdTe
CdTe
Konig et al. [Science 318, 766 (2007)]
Example 4: 3-dimensional
Z2 topological insulator
3-dimensional Z2 topological insulator
Moore & Balents; Roy; Fu, Kane & Mele (2006, 2007)
Z2 topological insulator
surface Dirac fermion
bulk: band insulator
surface: an odd number of surface Dirac modes
characterized by Z2 topological numbers
bulk
insulator
Ex: tight-binding model with SO int. on the diamond lattice
[Fu, Kane, & Mele; PRL 98, 106803 (2007)]
trivial insulator
Z2 topological
insulator
trivial band insulator:
0 or an even number of
surface Dirac modes
Surface Dirac fermions
topological
insulator
• A “half” of graphene
K
E
K’
K’
K
K
K’
ky
kx
• An odd number of Dirac fermions in 2 dimensions
cf. Nielsen-Ninomiya’s no-go theorem
Experiments
photon
• Angle-resolved photoemission spectroscopy (ARPES)
Bi1-xSbx
Hsieh et al., Nature 452, 970 (2008)
An odd (5) number of surface Dirac modes were observed.
p, E
Experiments II
Bi2Se3
“hydrogen atom” of top. ins.
a single Dirac cone
Xia et al.,
Nature Physics 5, 398 (2009)
ARPES experiment
Band calculations (theory)
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 of
topological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
Kitaev, arXiv:0901.2686
Zoo of topological insulators/superconductors
Classification of topological insulators/SCs
Topological insulators are stable against (weak) perturbations.
Random deformation of Hamiltonian
Natural framework: random matrix theory
(Wigner, Dyson, Altland & Zirnbauer)
Assume only basic discrete symmetries:
(1) time-reversal symmetry
TH *T 1  H
TRS =
(2) particle-hole symmetry

1
CH C   H
(3) TRS  PHS
PHS =
0 no TRS

+1 TRS with T  T (integer spin)
-1 TRS with T   T (half-odd integer spin)
T  is y
0 no PHS
+1 PHS with C   C (odd parity: p-wave)
-1 PHS with C   C (even parity: s-wave)
 chiral symmetry [sublattice symmetry (SLS)]
TCH TC   H
1
3  3  1  10
(2) particle-hole symmetry
px+ipy

1 
H  ck
2
ck

 ck 
hk   
 ck 
 x h*k x  hk
Bogoliubov-de Gennes
hk  k x x  k y y    k z
C   x  CT
dx2-y2+idxy

1 
H  ck 
2
ck 
 y h*k y  hk

 ck  
hk   
 ck  



hk   k x2  k y2  x  k y k y y   k z
C  i y  CT
10 random matrix ensembles
IQHE
Z2 TPI
px+ipy
dx2-y2+idxy
Examples of topological insulators in 2 spatial dimensions
Integer quantum Hall Effect
Z2 topological insulator (quantum spin Hall effect) also in 3D
Moore-Read Pfaffian state (spinless p+ip superconductor)
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)
BZ: Brillouin zone
• Classification of 2d Dirac fermions
13 classes (13=10+3)
AIII, CI, DIII
Bernard & LeClair (‘02)
AII, CII
• Anderson delocalization in 2d
nonlinear sigma models
Z2 topological term (2) or WZW term (3)
Topological distinction of ground states
deformed “Hamiltonian”
n empty
bands
m filled
bands
ky
map from BZ to Grassmannian
kx
 2 U m  n U mU n  
homotopy class
IQHE (2 dim.)
In classes AIII, BDI, CII, CI, DIII, Hamiltonian can be made off-diagonal.
Projection operator is also off-diagonal.
 3 U n  
topological insulators labeled by an integer



d 3k 
1
1
1
 q  

tr
q

q
q

q
q
 q


2
24

Discrete symmetries limit possible values of  q
Z2 insulators in CII (chiral symplectic)
The integer number  q 
# of surface Dirac (Majorana) fermions
bulk
insulator
surface
Dirac
fermion
(3+1)D 4-component Dirac Hamiltonian
 m
H k   k  x     m     
 k 
AII:  i y H * k i y  H  k 
chS
 q   sgn m 
1
2
mz 
TRS
DIII:  y  y H * k  y  y  H  k 
AIII:  y H k  y  H k 
 
k  

 m 
PHS
z
(3+1)D 8-component Dirac Hamiltonian
 0
H   
D
CI: Dk   i y  k  i 5    y  y k   im
D k   D k 
T
 q   sgn m  2
CII: Dk   k  m  D k 
mz 
z
D

0
1
2
 q  0
 i y D* k i y  D k 
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
13 classes (13=10+3)
AIII, CI, DIII
Bernard & LeClair (‘02)
AII, CII
• Anderson delocalization in 2d
nonlinear 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
Wegner, Efetov, Larkin, Hikami, ….
topological Z2 term
or WZW term
 2 M   Z 2
 3 M   Z
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 Table
Periodic table of topological insulators
Classification in any dimension
Kitaev, arXiv:0901.2686
Classification of
topological 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 fermions
to be found experimentally in solid-state devices
Andreev bound states in p-wave superfluids
Z2 T.I. + s-wave SC
Majorana bound state
3He-B
(Fu & Kane)

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