introduction to topological insulators and superconductors...topological (band) insulators...
TRANSCRIPT
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Introduction to topological insulators and superconductors
古崎 昭 (理化学研究所)
topological insulators (3d and 2d)September 16, 2009
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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
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絶縁体 Insulator
• 電気を流さない物質
絶縁体
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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
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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
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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
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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
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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
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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
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Example 1: Integer QHE
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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
xyR
1
BveF
Lorentz force
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Integer quantum Hall effect (von Klitzing 1980)
Hxy
xxQuantization of Hall conductance
h
eixy
2
807.258122e
h
exact, robust against disorder etc.
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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 kkAkd
i,
2
1 2
filled band
kkkyx
uukkA ,
integer valued
)(ruek
rki
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Edge states
• There is a gapless edge mode along the sample boundary.
B
Number of edge modes Che
xy
/2
Robust against disorder (chiral fermions cannot be backscattered)
Bulk: (2+1)d Chern-Simons theory
Edge: (1+1)d CFT
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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
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Example 2:chiral p-wave superconductor
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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 yx
d
Integrating outGinzburg-Landau
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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 : SSh
h
k
k
winding (wrapping)number=1
1zL
-
yxyxF
iik
im
H
22
22
rv
rue
tr
triEt /
,
,
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 ,
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Majorana edge state
v
uE
v
u
km
ik
i
ik
ikm
Fyxyx
F
yx
F
Fyx
222
222
2
1
2
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
k
ktxik
k
ktxikee
dktx
0
//
4,
yydyk
ik
kdkH y
F
k
kk
F
F
0
edge
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• Majorana bound state in a quantum vortex
vortex e
hc
Fn En / ,2
000
Bogoliubov-de Gennes equation
v
u
v
u
he
ehi
i
*
0
0 FEAepm
h 2
02
1
zero mode
v
u
00 00 Majorana (real) fermion!
energy spectrum of vortex bound states
2N vortices GS degeneracy = 2N
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interchanging vortices braid groups, non-Abelian statistics
i i+11 ii
ii 1
D.A. Ivanov, PRL (2001)
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Fractional quantum Hall effect at
• 2nd Landau level
• Even denominator (cf. Laughlin states: odd denominator)
• Moore-Read (Pfaffian) state
2
5
jjj iyxz
4/2
MR
21Pf i
z
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)
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Example 3: Z2 topological insulator
Quantum spin Hall effect
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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
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(trivial)Band insulator
Quantum SpinHall state
Quantum Hallstate
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kykx
E
K
K’
K’
K’
K
K
Kane-Mele model (PRL, 2005)
ij ij i
iiiv
ij
jzijiRj
z
iijSOji cccdscicscicctH
t
SOi
SOi
i j
ijd
A B
, ,, skke sBsArki
zvyxxyRzzSOyyxxK sssivHK 2
333 :
(B) 1 (A), 1 i
zvyxxyRzzSOyyxxK sssivHK 2
333 :' '
'
*
K
y
K
y HisHis time reversal symmetry
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• 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 modelgraphene + SOI[PRL 95, 146802 (2005)]
Quantum spin Hall effect
2
esxy
(if Sz is conserved)
0R
Edge states stable against disorder (and interactions)
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Z2 topological number
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Z2: stability of gapless edge states
(1) A single Kramers doublet
zzyy
x
xx
z sVsVsVVivsH 0
HisHis yy *
(2) Two Kramers doublets
zzyyxxyxz sVsVsVVsIivH 0
Kramers’ theorem
opens a gap
Odd number of Kramers doublet (1)
Even number of Kramers doublet (2)
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ExperimentHgTe/(Hg,Cd)Te quantum wells
Konig et al. [Science 318, 766 (2007)]
CdTeHgCdTeCdTe
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Example 4: 3-dimensionalZ2 topological insulator
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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
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Surface Dirac fermions
• A “half” of graphene
• An odd number of Dirac fermions in 2 dimensions
cf. Nielsen-Ninomiya’s no-go theorem
kykx
E
K
K’
K’
K’
K
K
topologicalinsulator
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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.
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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
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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)
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Classification oftopological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
Kitaev, arXiv:0901.2686
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Zoo of topological insulators/superconductors
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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
TT (integer spin)
(half-odd integer spin)
(2) particle-hole symmetry
HCCH 10 no PHS
PHS = +1 PHS with-1 PHS with
CC
CC (odd parity: p-wave)
(even parity: s-wave)
HTCTCH 1 10133
(3) TRS PHS chiral symmetry [sublattice symmetry (SLS)]
yisT
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(2) particle-hole symmetry Bogoliubov-de Gennes
zkyyxxkk
k
kkk kkhc
chccH
2
1
px+ipy
T
xkxkx CChh *
dx2-y2+idxy
zkyyyxyxkk
k
kkkkkkkh
c
chccH
22 2
1
T
ykyky CiChh *
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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
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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.
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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
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Topological distinction of ground states
m filledbands
n emptybands
xk
yk
map from BZ to Grassmannian
nUmUnmU2 IQHE (2 dim.)
homotopy class
deformed “Hamiltonian”
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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
qqqqqq
kdq
1112
3
tr24
Discrete symmetries limit possible values of q
Z2 insulators in CII (chiral symplectic)
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The integer number # of surface Dirac (Majorana) fermions q
bulkinsulator
surfaceDiracfermion
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(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
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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)
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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
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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.
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Reordered TableKitaev, arXiv:0901.2686
Periodic table of topological insulators
Classification in any dimension
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Classification oftopological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
Kitaev, arXiv:0901.2686
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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
Andreev bound states in p-wave superfluids 3He-B
Z2 T.I. + s-wave SC Majorana bound state (Fu & Kane)