トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy...
Transcript of トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy...
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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
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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
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
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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
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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
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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
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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
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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
Topological superconductorsBCS超伝導体
(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
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
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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.
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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
mceBnE ccn ωωh
AlGaAs GaAsBr
cyclotron motion
k
E
ππ−
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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
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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
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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
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( )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
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Example 2: chiral p-wave superconductor
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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
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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
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( ) ( )[ ]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=
∂∂
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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
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• 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 −→
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interchanging vortices braid groups, non-Abelian statistics
i i+11+→ ii γγ
ii γγ −→+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
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)
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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)
Br
Br
−
up-spin electrons
down-spin electrons
σλ rr⋅L
No spin rotation symmetry
![Page 28: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/28.jpg)
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
![Page 29: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/29.jpg)
• 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)
![Page 30: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/30.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 ++⊗++∂⊗= τ0
remain to be gapless
opens a gap
Odd number of Kramers doublet (1)
Even number of Kramers doublet (2)
![Page 31: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/31.jpg)
ExperimentHgTe/(Hg,Cd)Te quantum wells
Konig et al. [Science 318, 766 (2007)]
CdTeHgCdTeCdTe
![Page 32: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/32.jpg)
Example 4: 3-dimensional Z2
topological insulator
![Page 33: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/33.jpg)
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
![Page 34: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/34.jpg)
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
![Page 35: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/35.jpg)
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.
![Page 36: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/36.jpg)
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
![Page 37: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/37.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 38: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/38.jpg)
Classification oftopological insulators/superconductors
Schnyder, Ryu, AF, and Ludwig, PRB (2008)
Kitaev, arXiv:0901.2686
![Page 39: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/39.jpg)
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Zoo of topological insulators/superconductors
![Page 41: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/41.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 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 =
![Page 42: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/42.jpg)
(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 *
![Page 43: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/43.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 44: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/44.jpg)
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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.
![Page 46: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/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 2dnonlinear sigma models
Z2
topological term (2)
or
WZW term (3)
BZ: Brillouin zone
![Page 47: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/47.jpg)
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”
![Page 48: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/48.jpg)
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)
![Page 49: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/49.jpg)
The integer number # of surface Dirac (Majorana) fermions( )qν
bulkinsulator
surfaceDiracfermion
![Page 50: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/50.jpg)
(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
![Page 51: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/51.jpg)
(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
![Page 52: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/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 2dnonlinear sigma models
Z2
topological term (2)
or
WZW term (3)
![Page 53: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/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
termor WZW term
Wegner, Efetov, Larkin, Hikami, ….
( ) 22 ZM =π
( ) ZM =3π
![Page 54: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/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: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/55.jpg)
Reordered TableKitaev, arXiv:0901.2686
Periodic table of topological insulators
Classification in any dimension
![Page 56: トポロジカル絶縁体・超伝導体qft.web/2009/slides/...2 σ ∈Ζ× Edge states GS σ xy =−2 σ xy =−1 σ xy =0 σ xy =+1 σ xy =+2 Quantum phase transitions Space of](https://reader036.fdocument.pub/reader036/viewer/2022081402/5f1227c437e09017520f4084/html5/thumbnails/56.jpg)
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
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