Topological Insulators
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Transcript Topological Insulators
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
2i filled band
k y k x k x k y
1
2
d k k Ak x , k y
Ak x , k y uk k uk
2i
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
mx
Domain wall fermion
H iv x x y y mx z
x
x
1
1
x, y expiky y mx'dx'
0
v
i
E vk
m0
m0
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 / ur
e
r , t
vr
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
y0
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 n0 , 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 iSO c s c j iR 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
iSO
A
j
i
B
e
ik r
ky
kx
K’
K
iSO
E
K’
A, s
k
B ,s
k
s ,
3
K : H K iv x y 3 3SO s R y s x x s y v z
2
3
x
y
z z
K ': H K ' iv x y 3 3SO 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
ck
ck
hk
ck
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
ck
y h*k y hk
ck
hk
ck
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 mU 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
mz
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: Dk i y k i 5 y y k im
D k D k
T
q sgn m 2
CII: Dk k m D k
mz
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)