Introduction to topological insulators and superconductors
古崎 昭 (理化学研究所)
topological insulators (3d and 2d)September 16, 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
2
2
2
rVarV
Periodic electrostatic potential from ions and other electrons (mean-field)
Bloch’s theorem:
ruarua
ka
ruer knknknikr
,,, , ,
kEn Energy band dispersion :n band index
例:一次元格子
2
A A A A AB B B B B
t
n 1n1n
B
Aikn
u
uen
B
A
B
A
u
uE
u
u
kt
kt
)2/cos(2
)2/cos(2
222 )2/(cos4 ktE
k
E
0
2
Metal and insulator in the band theory
k
E
a
a 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
a
a 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.
2
2
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
free fermions
Topological (band) insulators
Condensed-matter realization of domain wall fermions
Examples:
• 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 superconductors
BCS 超伝導体 超伝導 gap
(Dirac or Majorana) をもつ
p+ip superconductor, 3He
stable
Example 1: Integer QHE
Prominent example: quantum Hall effect• Classical Hall effect
B
e
E
v
W
:n electron density
Electric current nevWI
Bc
vE Electric field
Hall voltage Ine
BEWVH
Hall resistancene
BRH
Hall conductanceH
xy R
1BveF
Lorentz force
Integer quantum Hall effect (von Klitzing 1980)
Hxy
xx Quantization of Hall conductance
h
eixy
2
807.258122e
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)
Landau levels:
,...2,1,0 , ,21 n
mc
eBnE ccn
AlGaAs GaAsB
cyclotron motion
k
E
TKNN number (Thouless-Kohmoto-Nightingale-den Nijs)TKNN (1982); Kohmoto (1985)
Ch
exy
2
Chern number (topological invariant)
yxxy k
u
k
u
k
u
k
urdkd
iC
**22
2
1
yxk kkAkdi
, 2
1 2
filled band
kkkyx uukkA
,
integer valued
)(ruek
rki
Edge states• There is a gapless edge mode along the sample
boundary.B
Number of edge modes Chexy
/2
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
zyyxx mivH
parity anomaly mxy sgn2
1
Domain wall fermion
i
dxxmv
ikyyxx
y
1''
1exp,
0
vkE 0m 0m
Example 2: chiral p-wave superconductor
BCS 理論 (平均場理論)
,,
*,,
,
22
'','','2
1
2
rrrrrrrrrrdd
rAiem
rrdH
dd
d
''',, rrrrVrr 超伝導秩序変数
S-wave (singlet) rrrrr 2
1'',,
0
P-wave (triplet)
+
rrrr
rr
''
',,
yxkk ikkk 0)(
高温超伝導体 22)( yxkk kkk
22 yxd
Integrating out Ginzburg-Landau
Spinless px+ipy superconductor in 2 dim.• Order parameter
• Chiral (Majorana) edge state
)()( 0 yxkk ikkk
k
E
2
1
2)()(
)(2)(
2
122
22
k
FFyx
FyxF
kh
mkkkikk
kikkmkkH
Hamiltonian density
k
k
m
kkk
k
k
k
kh Fyx
F
y
F
xk 2
222
khE
Gapped fermion spectrum22 : SS
h
h
k
k
winding (wrapping)number=1
1zL
yxyxF
iik
im
H
222
2
rv
rue
tr
tr iEt /
,
,
v
uE
v
u
hii
iih
yx
yx
0
0
Hamiltonian density
Bogoliubov-de Gennes equation
*
*
,u
v
v
uEE Particle-hole symmetry (charge conjugation)
zeromode: Majorana fermion
Ht
i ,
Majorana edge state
v
uE
v
u
km
ik
i
ik
ikm
FyxyxF
yxF
Fyx
222
222
2
12
1
E
px+ipy superconductor: 0y 0
v
u
y
1
1 cosexp 22 ykky
k
mikx
v
uF
Fk
k
Fk
kE
k
E
2
F
FF
k
kktxik
kktxik ee
dktx
0
//
4,
yydyk
ik
kdkH y
F
k
kkF
F
0
edge
• Majorana bound state in a quantum vortex
vortex e
hc
Fn En / , 2000
Bogoliubov-de Gennes equation
v
u
v
u
he
ehi
i
*0
0 FEAepm
h 2
0 2
1
zero mode
v
u
00 00 Majorana (real) fermion!
energy spectrum of vortex bound states
2N vortices GS degeneracy = 2N
interchanging vortices braid groups, non-Abelian statistics
i i+11 ii
ii 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
2
5
jjj iyxz
4/2MR
21Pf iz
jiji
ji
ezzzz
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)
B
B
up-spin electrons
down-spin electrons
SEpSL
,,,spin
,,charge
2
0
xyxyxyxy
xyxyxy
If Sz is conserved, If Sz is NOT conserved,
Chern # (Z) Z2
Quantized spin Hall conductivity
(trivial)Band insulator
Quantum SpinHall state
Quantum Hallstate
ky
kx
E
K
K’
K’
K’
K
K
Kane-Mele model (PRL, 2005)
ij ij i
iiivij
jzijiRjz
iijSOji cccdscicscicctH
t
SOi
SOi
i j
ijd
A B , ,, skke sBsA
rki
zv
yxxyR
zzSOy
yx
xK sssivHK
2
333 :
(B) 1 (A), 1 i
zv
yxxyR
zzSOy
yx
xK sssivHK
2
333 :' '
'*
Ky
Ky HisHis time reversal symmetry
• Quantum spin Hall insulator is characterized by Z2 topological index
1 an odd number of helical edge modes; Z2 topological insulator
an even (0) number of helical edge modes01 0
Kane-Mele model graphene + SOI [PRL 95, 146802 (2005)]
Quantum spin Hall effect
2
esxy
(if Sz is conserved)
0R
Edge states stable against disorder (and interactions)
Z2 topological number
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 0
Kramers’ theorem
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 fermionZ2 topological insulator
bulk: band insulator
surface: 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 dimensions cf. Nielsen-Ninomiya’s no-go
theorem
ky
kx
E
K
K’
K’
K’
K
K
topologicalinsulator
Experiments• Angle-resolved photoemission spectroscopy (ARPES)
Bi1-xSbx
p, Ephoton
Hsieh et al., Nature 452, 970 (2008)
An odd (5) number of surface Dirac modes were observed.
Experiments II
Bi2Se3
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 TRSTRS = +1 TRS with -1 TRS with
TT
TT (integer spin)(half-odd integer spin)
(2) particle-hole symmetry
HCCH 1 0 no PHSPHS = +1 PHS with -1 PHS with
CC
CC (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
c
chccH
2
1px+ipy
Txkxkx CChh *
dx2-y2+idxy
zkyyyxyxkk
kkkk
kkkkhc
chccH
22 2
1
Tykyky CiChh *
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 2d nonlinear sigma models Z2 topological term (2) or WZW term (3)
BZ: Brillouin zone
Topological distinction of ground states
m filledbands
n emptybands
xk
yk
map from BZ to Grassmannian
nUmUnmU2 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.
nU3 topological insulators labeled by an integer
qqqqqqkd
q
111
2
3
tr24
Discrete symmetries limit possible values of q
Z2 insulators in CII (chiral symplectic)
The integer number # of surface Dirac (Majorana) fermions q
bulkinsulator
surfaceDiracfermion
(3+1)D 4-component Dirac Hamiltonian
mk
kmmkkH x
AII: kHikHi yy *TRS
DIII: kHkH yyyy *PHS
AIII: kHkH yy chS
mq sgn2
1
zm
z
(3+1)D 8-component Dirac Hamiltonian
0
0
D
DH
imkikikD yyy 5CI:
kDkDT
CII: kDmkkD kDikDi yy *
2sgn2
1 mq
0q
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 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 topological Z2 term
or WZW term
Wegner, Efetov, Larkin, Hikami, ….
22 ZM
ZM 3
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 fermions to be found experimentally in solid-state devices
Andreev bound states in p-wave superfluids 3He-B
Z2 T.I. + s-wave SC Majorana bound state (Fu & Kane)