introduction to topological insulators and superconductors 古崎 昭 (理化学研究所)...
TRANSCRIPT
![Page 1: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/1.jpg)
Introduction to topological insulators and superconductors
古崎 昭 (理化学研究所)
topological insulators (3d and 2d)September 16, 2009
![Page 2: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/2.jpg)
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
![Page 3: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/3.jpg)
絶縁体 Insulator
• 電気を流さない物質
絶縁体
![Page 4: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/4.jpg)
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
![Page 5: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/5.jpg)
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
![Page 6: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/6.jpg)
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
![Page 7: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/7.jpg)
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
![Page 8: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/8.jpg)
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
![Page 9: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/9.jpg)
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
![Page 10: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/10.jpg)
Example 1: Integer QHE
![Page 11: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/11.jpg)
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
![Page 12: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/12.jpg)
Integer quantum Hall effect (von Klitzing 1980)
Hxy
xx Quantization of Hall conductance
h
eixy
2
807.258122e
h
exact, robust against disorder etc.
![Page 13: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/13.jpg)
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
![Page 14: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/14.jpg)
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
![Page 15: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/15.jpg)
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
![Page 16: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/16.jpg)
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
![Page 17: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/17.jpg)
Example 2: chiral p-wave superconductor
![Page 18: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/18.jpg)
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
![Page 19: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/19.jpg)
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
![Page 20: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/20.jpg)
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 ,
![Page 21: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/21.jpg)
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
![Page 22: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/22.jpg)
• 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
![Page 23: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/23.jpg)
interchanging vortices braid groups, non-Abelian statistics
i i+11 ii
ii 1
D.A. Ivanov, PRL (2001)
![Page 24: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/24.jpg)
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)
![Page 25: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/25.jpg)
Example 3: Z2 topological insulator
Quantum spin Hall effect
![Page 26: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/26.jpg)
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
![Page 27: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/27.jpg)
(trivial)Band insulator
Quantum SpinHall state
Quantum Hallstate
![Page 28: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/28.jpg)
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
![Page 29: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/29.jpg)
• 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)
![Page 30: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/30.jpg)
Z2 topological number
![Page 31: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/31.jpg)
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)
![Page 32: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/32.jpg)
ExperimentHgTe/(Hg,Cd)Te quantum wells
Konig et al. [Science 318, 766 (2007)]
CdTeHgCdTeCdTe
![Page 33: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/33.jpg)
Example 4: 3-dimensional Z2 topological insulator
![Page 34: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/34.jpg)
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
![Page 35: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/35.jpg)
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
![Page 36: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/36.jpg)
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.
![Page 37: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/37.jpg)
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
![Page 38: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/38.jpg)
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)
![Page 39: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/39.jpg)
Classification oftopological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
Kitaev, arXiv:0901.2686
![Page 40: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/40.jpg)
![Page 41: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/41.jpg)
Zoo of topological insulators/superconductors
![Page 42: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/42.jpg)
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
![Page 43: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/43.jpg)
(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 *
![Page 44: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/44.jpg)
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
![Page 45: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/45.jpg)
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.
![Page 46: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/46.jpg)
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
![Page 47: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/47.jpg)
Topological distinction of ground states
m filledbands
n emptybands
xk
yk
map from BZ to Grassmannian
nUmUnmU2 IQHE (2 dim.)
homotopy class
deformed “Hamiltonian”
![Page 48: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/48.jpg)
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)
![Page 49: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/49.jpg)
The integer number # of surface Dirac (Majorana) fermions q
bulkinsulator
surfaceDiracfermion
![Page 50: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/50.jpg)
(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
![Page 51: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/51.jpg)
(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
![Page 52: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/52.jpg)
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)
![Page 53: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/53.jpg)
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
![Page 54: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/54.jpg)
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.
![Page 55: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/55.jpg)
Reordered TableKitaev, arXiv:0901.2686
Periodic table of topological insulators
Classification in any dimension
![Page 56: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/56.jpg)
Classification oftopological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
Kitaev, arXiv:0901.2686
![Page 57: Introduction to topological insulators and superconductors 古崎 昭 (理化学研究所) topological insulators (3d and 2d) September 16, 2009](https://reader035.vdocuments.mx/reader035/viewer/2022081513/56649c765503460f94929e39/html5/thumbnails/57.jpg)
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)